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PUTTING AUCTION THEORY TO WORK
This book provides a comprehensive introduction to modern auction theory and its important new applications. It is written by a leading economic theorist whose suggestions guided the creation of the new spectrum auction designs. Aimed at graduate students and professionals in economics, the book gives the most up-to-date treatments of both traditional theories of “optimal auctions” and newer theories of multi-unit auctions and package auctions, and shows by example how these theories are used. The analysis explores the limitations of prominent older designs, such as the Vickrey auction design, and evaluates the practical responses to those limitations. It explores the tension between the traditional theory of auctions with a fixed set of bidders, in which the seller seeks to squeeze as much revenue as possible from the fixed set, and the theory of auctions with endogenous entry, in which bidder profits must be respected to encourage participation. It shows how seemingly different auction designs can lead to nearly identical outcomes if the participating bidders are the same – a finding that focuses attention on (1) attracting bidders and (2) mini- mizing the cost of running the auction and bidding in it. It shows how new auc- tion designs can accommodate complicated procurement settings and sales with many interrelated goods.
Paul Milgrom is Leonard and Shirley Ely Professor of Humanities and Social Sciences and Professor of Economics at Stanford University.He has also taught at Harvard University and MIT. A Fellow of the American Academy of Arts and Sciences and the Econometric Society, Professor Milgrom has served on the editorial boards of the American Economic Review, Ecometrica, the Journal of Economic Theory, the Journal of Economics and Management Strategy, and Games and Economic Behavior. He is coauthor with John Roberts of the 1992 landmark text Economics, Organization, and Management. Professor Milgrom’s research has been published in the leading journals in economics, including the American Economic Review, Econometrica, the Journal of Political Economy, the Quarterly Journal of Economics, the Journal of Economic Theory, the Journal of Economic Perspectives, and the Journal of MathematicalEconomics.Hiscurrentresearchinterestsareinincentivetheory, planning, and auction market design. Professor Milgrom is internationally known for his work in spectrum auction designs.
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CHURCHILL LECTURES IN ECONOMICS
The Churchill Lectures in Economics was inaugurated in 1993 to provide a se- ries of public lectures on topics of current interest to students and researchers in the discipline. The lectures will be selected from the top echelon of leading scholars in the profession. Although they will always be acknowledged special- ists in their field, they will be encouraged to take a broad look at their chosen subject and to reflect in a way that will be accessible to senior undergraduates and graduate students.
Peter Diamond, On Time, 1994 Douglas Gale, Strategic Foundations of General Equilibrium: Dynamic Match- ing and Bargaining Games, 2000 Ariel Rubinstein, Economics and Language, 2000
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PUTTING AUCTION THEORY TO WORK
PAUL MILGROM Stanford University
v cambridge university press Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo
Cambridge University Press The Edinburgh Building, Cambridge cb2 2ru, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521551847
© Paul Milgrom 2004
This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press.
First published in print format 2004 isbn-13 978-0-511-16612-9 eBook (NetLibrary) isbn-10 0-511-16612-5 eBook (NetLibrary) isbn-13 978-0-521-55184-7 hardback isbn-10 0-521-55184-6 hardback isbn-13 978-0-521-53672-1 paperback isbn-10 0-521-53672-3 paperback
Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. P1: FCH/FFX P2: FCH/FFX QC: FCH/FFX T1: FCH CB610-FM CB610-Milgrom-v3 October 27, 2003 16:27
Contents
Preface page xi Foreword by Evan Kwerel xv 1 Getting to Work 1 1.1 Politics Sets the Stage3 1.2 Designing for Multiple Goals3 1.2.1 Substitutes and Complements6 1.2.2 New Zealand’s Rights Auction9 1.2.3 Better Auction Designs 13 1.2.4 The FCC Design and Its Progeny 13 1.3 Comparing Seller Revenues 16 1.4 The Academic Critics 19 1.4.1 Resale and the Coase Theorem 19 1.4.2 Mechanism Design Theory 21 1.4.3 Theory and Experiment 25 1.4.4 Practical Concerns 26 1.5 Plan for This Book 31
PART I THE MECHANISM DESIGN APPROACH 35 2 Vickrey–Clarke–Groves Mechanisms 45 2.1 Formulation 45 2.2 Always Optimal and Weakly Dominant Strategies 49 2.3 Balancing the Budget 53 2.4 Uniqueness 55 2.5 Disadvantages of the Vickrey Auction 56 2.5.1 Practical Disadvantages 56 2.5.2 Monotonicity Problems 57 2.5.3 The Merger–Investment Disadvantage 60 2.6 Conclusion 61
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3 The Envelope Theorem and Payoff Equivalence 64 3.1 Hotelling’s Lemma 65 3.2 The Envelope Theorem in Integral Form 66 3.3 Quasi-linear Payoffs 69 3.3.1 Holmstrom’s Lemma 70 3.3.2 The Green–Laffont–Holmstrom Theorem 71 3.3.3 Myerson’s Lemma 73 3.3.4 Revenue Equivalence Theorems 75 3.3.5 The Myerson–Satterthwaite Theorem 77 3.3.6 The Jehiel–Moldovanu Impossibility Theorems 80 3.3.7 Myerson and Riley–Samuelson Revenue-Maximizing Auctions 84 3.3.8 The McAfee–McMillan Weak-Cartels Theorem 87 3.3.9 Sequential Auctions and Weber’s Martingale Theorem 90 3.3.10 Matthews Theorem: Risk Averse Payoff Equivalence 91 3.4 Conclusion 94
4 Bidding Equilibrium and Revenue Differences 98 4.1 The Single Crossing Conditions 99 4.1.1 The Monotonic Selection Theorem 101 4.1.2 The Sufficiency Theorem 102 4.1.3 The Constraint Simplification Theorem 105 4.1.4 The Mirrlees–Spence Representation Theorem 106 4.2 Deriving and Verifying Equilibrium Strategies 110 4.2.1 The Second-Price Auction with a Reserve Price 111 4.2.2 The Sealed Tender, or First-Price, Auction 112 4.2.3 The War of Attrition Auction 117 4.2.4 The All-Pay Auction 119 4.3 Revenue Comparisons in the Benchmark Model 119 4.3.1 Payoff Equivalence without Revenue Equivalence 121 4.3.2 Budget Constraints 132 4.3.3 Endogenous Quantities 135 4.3.4 Correlated Types 137 4.4 Expected-Revenue-Maximizing Auctions 140 4.4.1 Myerson’s Theorem 144 4.4.2 Bulow–Klemperer Theorem 148 4.4.3 The Irregular Case 148 4.5 Auctions with Weak and Strong Bidders 149 4.6 Conclusion 154 P1: FCH/FFX P2: FCH/FFX QC: FCH/FFX T1: FCH CB610-FM CB610-Milgrom-v3 October 27, 2003 16:27
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5 Interdependence of Types and Values 157 5.1 Which Models and Assumptions are “Useful”? 158 5.1.1 Payoffs Depend Only on Bids and Types 158 5.1.2 Types Are One-Dimensional and Values Are Private 159 5.1.3 Types Are Statistically Independent 161 5.2 Statistical Dependence and Revenue-Maximizing Auctions 162 5.3 Wilson’s Drainage Tract Model 166 5.3.1 Equilibrium 167 5.3.2 Profits and Revenues 173 5.3.3 Bidder Information Policy 175 5.3.4 Seller Information Policy 177 5.4 Correlated Types and Interdependent Values 181 5.4.1 Affiliation 182 5.4.2 The Milgrom–Weber Ascending Auction Models 187 5.4.2.1 The (Second-Price) Button Auction with Minimal Information 188 5.4.2.2 The Button Auction with Maximal Information 195 5.4.2.3 Some Revenue Comparisons 198 5.4.3 First-Price Auctions 200 5.5 Conclusion 204
6 Auctions in Context 208 6.1 The Profit and Surplus Contribution of an Entrant 214 6.2 Symmetric Models with Costly Entry 216 6.2.1 Symmetric Bidders and Uncoordinated Entry 218 6.2.1.1 Equilibrium in Entry and Bidding Decisions 218 6.2.1.2 Setting the Reserve Price 222 6.2.2 Coordinating Entry among Symmetric Competitors 225 6.2.2.1 Pre-qualifying Bidders 227 6.2.2.2 Auctions, Negotiations, and Posted Prices 230 6.2.2.3 Buy Prices 232 6.3 Asymmetric Models: Devices to Promote Competition 234 6.3.1 Example of Set-asides 235 6.3.2 Example of Bidding Credits 237 6.3.3 Example of Lot Structure and Consolation Prizes 238 6.3.4 Premium Auctions 239 6.3.5 Dutch vs. English Auctions and the Anglo-Dutch Design 241 6.4 After the Bidding Ends 243 6.4.1 Bankruptcy and Non-performance 243 6.4.2 Scoring Rules vs. Price-Only Bids 245 6.5 Conclusion 247 P1: FCH/FFX P2: FCH/FFX QC: FCH/FFX T1: FCH CB610-FM CB610-Milgrom-v3 October 27, 2003 16:27
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PART II MULTI-UNIT AUCTIONS 251 7 Uniform Price Auctions 255 7.1 Uniform Price Sealed-Bid Auctions 257 7.1.1 Demand Reduction 258 7.1.2 Low-Price Equilibria 262 7.2 Simultaneous Ascending Auctions 265 7.2.1 The Simultaneous Ascending Auction and the Walrasian Tatonnement 268 7.2.2 Clock Auctions 279 7.2.3 Strategic Incentives in Uniform Price Auctions 284 7.2.3.1 The Basic Clock Auction Model 284 7.2.3.2 The Alternating-Move Clock Auction 287 7.2.3.3 Strategic Incentives with Elastic Supply 290 7.3 Conclusion 293 8 Package Auctions and Combinatorial Bidding 296 8.1 Vickrey Auctions and the Monotonicity Problems 302 8.1.1 Bidders’ Vickrey Payoffs Bound Their Core Payoffs 305 8.1.2 Vickrey Auctions and the Entry Puzzle 305 8.1.3 When Are Vickrey Outcomes in the Core? 307 8.1.4 Substitute Goods and Core Outcomes 308 8.1.5 Substitute Goods and Vickrey Outcomes 312 8.2 Bernheim–Whinston First-Price Package Auctions 315 8.2.1 Formulation 316 8.2.2 Profit-Target Strategies 318 8.2.3 Equilibrium and the Core 319 8.3 Ausubel–Milgrom Ascending Proxy Auctions 324 8.3.1 The Proxy Auction with Unlimited Budgets 325 8.3.1.1 Proxy Outcomes Are Core Outcomes 326 8.3.1.2 Profit-Target Strategies and Equilibrium 327 8.3.1.3 The Proxy Auction When Goods Are Substitutes 329 8.3.2 The Non-transferable-Utility Proxy Auction 330 8.4 Conclusion 333
Bibliography 339 Author Index 347 Subject Index 351 P1: FCH/FFX P2: FCH/FFX QC: FCH/FFX T1: FCH CB610B-PRE CB610-Milgrom-v3 October 10, 2003 11:11
Preface
This book synthesizes the insights I have found from my teaching, research, and consulting about auction design. For me, the three have long been intertwined. I wrote my Ph.D. thesis about auction theory under the guidance of Robert Wilson, who was then already advising biddersabouthowtobidandgovernmentsabouthowtodesignauctions. Fifteen years later, Wilson and I together made proposals that became the basis for the design of the Federal Communications Commission (FCC) spectrum auctions – the most influential new auction design of the twentieth century. The FCC design was copied with variations for spectrum sales on six continents. In the intervening years, I had often taught about auction theory, though not yet as the practical subject that it was to become. Work on this book began in spring of 1995, when I delivered the Churchill lectures at Cambridge University. Those lectures emphasized the history and design of the spectrum auctions run by the FCC begin- ning in 1994, as well as the bidders’ experiences in the auctions. Wilson and I had only a few weeks in which to form our design and make rec- ommendations, and my “Churchill project” was to complete the analysis of those recommendations by identifying the kinds of environments in which our new design was likely to be effective. Events caused the project to be delayed, but the project received a boost and a twist when I de- livered lectures about auction theory in courses at Stanford in 1996 and 2000, in Jerusalem in 1997, and at Harvard and MIT in 2001 and 2002. In my 1978 dissertation, I had written that there were seven main re- sults of auction theory. Two decades later, there are many more and many views about what is most important and how best to synthe- size this exceptionally beautiful theory. What is distinctive about my
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synthesis here and what makes it both more encompassing and more practical than earlier attempts is that it is rooted both in traditional de- mand theory and in real-world experiences.1 I unify auction theory with demand theory partly by using familiar techniques and concepts: the envelope theorem, comparative statics methods, and demand theory concepts like substitutes and complements. My perspectives on auction theory differ in emphasis and method fromthoseofseveralrecentcontributors.Inchapter1,Idescribehowone can use the stylized results of auction theory in practical design. Chap- ter 2 presents my distinctive treatment of the Vickrey auction, which explains how the striking theoretical advantages of the auction are offset by equally striking disadvantages, which too often go unremarked. Chapters 3 and 4 develop the classical results of auction theory using the tools of ordinary demand theory: the envelope theorem and the comparative statics techniques. This is in sharp contrast to graduate microeconomics textbooks that emphasize the distinctive “revelation principle” as the basic tool of mechanism design theory (Mas Colell, Whinston, and Green (1995)) – a tool that has no analog in or relevance for demand theory. In chapter 5, I revisit the models of auctions with interdependent values and correlated information to recast them in the same terms. These new treatments show that parts of auction theory that had seemed difficult can be treated simply by using the same methods. My experience in auction consulting teaches that clever new designs are only very occasionally among the main keys to an auction’s success. Much more often, the keys are to keep the costs of bidding low,encourage the right bidders to participate, ensure the integrity of the process, and take care that the winning bidder is someone who will pay or deliver as promised. Chapter 6 emphasizes those considerations. It particularly emphasizestheconsequencesoffreeentryandtheinstrumentsavailable to the designer to encourage entry of the right kinds. Chapters 7 and 8 deal with an area of auction design in which schol- arly input can add enormous value. This is in the area of multi-unit
1 In the years after the first FCC auctions, I contributed to spectrum auction designs in the United States, Germany, Australia, and Canada, electricity auction designs in New Jersey and Texas, asset sales in the United States and Mexico, and internet procurement auctions. My suggestions were also the principal basis of the FCC’s design for auction #31 – its first package or “combinatorial” auction design. P1: FCH/FFX P2: FCH/FFX QC: FCH/FFX T1: FCH CB610B-PRE CB610-Milgrom-v3 October 10, 2003 11:11
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auctions. Such auctions have been used for radio spectrum, electri- cal power, Treasury bills, and other applications. The design problems for these auctions include not just the usual ones about getting incen- tives and allocations right, but also limiting the complexity so that costs incurred by bidders are not too high and the reliability of the system is maintained. Unlike auctions for a single object, in which efficiency and revenue objectives are usually at least roughly aligned, multi-item auctions can involve radical trade-offs between these two objectives. Chapter 8, especially, highlights such trade-offs and explains how the new Ausubel–Milgrom design tries to reach a practical compromise. I owe debts to many people not only for their help in preparing this book, but for helping me to reach this point in my understanding of auc- tions. Robert Wilson introduced me to auction theory in graduate school, directed my Ph.D. research, and joined me in the work of creating the FCC auction for our joint client, Pacific Bell. I have dedicated this book to him. The folks at Pacific Bell, particularly James Tuthill, had the patience and courage to support my applied research and to help me advocate it to the FCC. Evan Kwerel and the FCC team repeatedly showed the courage to be innovators, trying out radical new ideas. The colleagues with whom I have consulted on auction designs – Larry Ausubel, Peter Cramton, Pre- ston McAfee, John McMillan, Charles Plott, and again Robert Wilson – inspired me with their ideas, enthusiasm, and inspiration. Many people have directly supported my efforts in writing this book. I am especially grateful to five students and colleagues who read the en- tire manuscript and made helpful suggestions. Professor Valter Sorana’s detailed and very thoughtful comments are reflected throughout the book. My research assistant, Hui Li, often sat next to me at my computer, insisting that certain passages or arguments needed further detail and prodding me to make the text, as she would say, “easy enough for me.” The Harvard graduate students Parag Pathak and Siva Anantham and the Stanford graduate student Paul Riskind all read the entire manuscript and made hundreds of suggestions. The undergraduate Dan Kinnamon read and commented on parts of the manuscript and provided research assistance for the buy-price model of chapter 6. I also had invaluable discussions about particular parts of the subject matter with many colleagues, including Susan Athey, Larry Ausubel, Jeremy Bulow, Peter Cramton, Paul Klemperer, Evan Kwerel, Benny Moldovanu, Noam Nisan, Motty Perry, Leo Rezende, John Roberts, Al Roth, David Salant, Ilya Segal, P1: FCH/FFX P2: FCH/FFX QC: FCH/FFX T1: FCH CB610B-PRE CB610-Milgrom-v3 October 10, 2003 11:11
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Padmanhabhan Srinagesh, Steve Tadelis, Bob Wilson, Lixin Ye, and Charles Zheng. The period since I began this work was an especially difficult one for me personally and for my family, and I thank them, too. Without the love and support of my wife, Eva Meyersson Milgrom, and my children, Joshua and Elana, I could not have finished this book. P1: FCH/FFX P2: FCH/FFX QC: FCH/FFX T1: FCH CB610B-FOR CB610-Milgrom-v3 October 10, 2003 11:9
Foreword
Paul Milgrom has had an enormous influence on the most important re- cent application of auction theory for the same reason you will want to read this book – clarity of thought and expression. In August 1993, Pres- ident Clinton signed legislation granting the Federal Communications Commission the authority to auction spectrum licenses and requiring it to begin the first auction within a year. With no prior auction experience and a tight deadline, the normal bureaucratic behavior would have been to adopt a “tried and true” auction design. In 1993, however, there was no tried and true method appropriate for the circumstances – multiple licenses with potentially highly interdependent values. I had been ad- vocating the use of auctions to select FCC licensees since 1983, when I joined the staff of the FCC’s Office of Plans and Policy. When auction legislation finally passed, I was given the task of developing an auction design. One of the first auction design issues the FCC considered was whether to use an ascending bid mechanism or a single round sealed bid. The federal government generally used sealed-bid auctions, especially for high-valued rights such as offshore oil and gas leases. FCC staff felt rea- sonably confident that we could implement a sealed-bid auction – keep thebidssecure,openthebids,andselectthehighbids.Thereweredoubts whether we could do anything more complex. In the end, the FCC chose an ascending bid mechanism, largely because we believed that provid- ing bidders with more information would likely increase efficiency and, as shown by Milgrom and Weber (1982a), mitigate the winner’s curse. The initial design the FCC proposed in September 1993 was a hybrid of an ascending bid and a first-price sealed-bid auction. It was intended to address the contentious policy issue of the appropriate geographic
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scope of the licenses for broadband personal communications services (PCS). Some companies argued that the FCC should issue nationwide licenses. Other companies, especially incumbent cellular providers that were barred from holding both a cellular and a PCS license in the same geographic area, argued for regional licenses. For each of two nation- wide spectrum blocks, the FCC proposed conducting a single round sealed-bid auction for all 51 licenses as a group, followed by a series of open outcry auctions for the same licenses individually. The sealed bids would be opened at the conclusion of the open outcry auctions, and the spectrum awarded to the highest sealed bid only if it exceeded the sum of bids on the individual licenses. The initial FCC proposal also discussed the possibility of a simul- taneous auction mechanism. Had AirTouch, a large cellular operator, not advocated this approach, it might not have been mentioned in the FCC’s September Notice of Proposed Rule Making. In a meeting with me, AirTouch pointed out that in my 1985 FCC working paper written with Lex Felker I had suggested a simplified system of simultaneous bid- ding where parties simultaneously placed independent bids on several licenses. In 1985 I had no idea how to run such a simultaneous auction, and in 1993 I was very skeptical of the possibility of anyone developing and the FCC implementing a workable simultaneous auction within the one year provided by the legislation; but Paul Milgrom and Bob Wilson (working for Pacific Bell) and Preston McAfee (working for AirTouch) completely changed my thinking. Both the Milgrom–Wilson and the McAfee propos- als were mindful of the limits on the complexity of any proposal that the FCC could or would implement. Both proposed simultaneous ascending bid auctions with discrete bidding rounds. This approach promised to provide much of the operational simplicity of sealed-bid auctions with the economic efficiency of an ascending auction. The 1993 legislation required that the FCC develop auction rules within 7 months and begin auctions within another 4 months. The FCC could have met the legislative mandate by beginning a sealed-bid auc- tion or an oral outcry auction. So why was it so important to begin a simultaneous auction within the legislative deadline? It was my view that whatever method was used in the first FCC auction, if it appeared successful, would become the default method for all future auctions, including broadband PCS. So I spent considerable effort looking for a P1: FCH/FFX P2: FCH/FFX QC: FCH/FFX T1: FCH CB610B-FOR CB610-Milgrom-v3 October 10, 2003 11:9
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set of licenses for our first auction that the FCC could successfully auc- tion using the simultaneous multiple round design. I proposed to senior FCC staff that we auction 10 narrowband PCS licenses. This was a small enough number that we could successfully implement a simultaneous auction, and the licenses were valuable enough that a success would be considered important, but not so valuable that a failure would impose an unacceptably large loss. The closing rule was one of the major design issues for a simultaneous auction. McAfee proposed a market-by-market closing rule with adjust- ments in bid increments to foster markets closing at approximately the same time. In contrast, Milgrom and Wilson proposed a simultaneous closing rule whereby the auction closes on all licenses only after a round has passed with no bidding on any license. Until then, bidding remains open on all licenses. McAfee proposed the market-by-market closing rule because of its operational simplicity. The FCC could surely run a number of separate ascending bid auctions in parallel. Milgrom argued however, that market-by-market closing could potentially foreclose ef- ficient backup strategies. (For example, you might be the high bidder on a license for several rounds while a license that is a substitute for you closed. If you were then outbid on your license, you would not have the opportunity to place a bid on the substitute.) Milgrom’s argument pre- vailed, and the FCC adopted a simultaneous closing rule, but not before addressing a closely related issue. Would an auction with the simultaneous closing rule proposed by Milgrom and Wilson ever end? This was the worst case scenario that troubled me when I first met Paul Milgrom. He had come to the FCC to explain their auction design. The simultaneous multiple round auction with a simultaneous closing rule struck me as the most elegant solution I had seen for auctioning multiple licenses that could be both substitutes and complements. But might bidders each have an incentive to hold back while observing the bids made by others? If so, how could the FCC be sure that the auction would close in a timely fashion? I asked Milgrom this question. He clearly had thought about the problem and responded that with no loss of efficiency, bidders could be required to be active on at least one license in every round. Any serious bidder must either have a high bid or place an acceptable new bid. With only 20 days between Comments and the deadline for Reply Comments, Milgrom and Wilson developed this insight into the activity rule that the FCC has used in all P1: FCH/FFX P2: FCH/FFX QC: FCH/FFX T1: FCH CB610B-FOR CB610-Milgrom-v3 October 10, 2003 11:9
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its simultaneous multiple round auctions. The Milgrom–Wilson activity rule was an elegant, novel solution to a difficult practical auction design issue. It imposed a cost on holding back by tying a bidder’s level of eligi- bility in future rounds to its activity level in the current round. If a bidder is not active on a minimum percentage of the quantity of spectrum for which it is eligible to bid, it suffers a permanent loss of eligibility. This discourages bidders from holding back, whether to “hide in the grass” or to collusively divide up the market. The activity rule was critical to the FCC adopting the Milgrom–Wilson auction design. The FCC could not tolerate the risk that the auction would drag on indefinitely with little bidding. The activity rule, with the ability to increase the activity requirement during the action, provided theFCCwithamechanismtopromoteareasonableauctionpacewithout subjecting bidders to the risk of an unanticipated close when they still wished to make additional bids. Without this feature the broadband PCS auction might have ended after only 12 rounds with revenue at 12% of the actual total. Because of less than anticipated initial eligibility in the auction, the initial level of the activity requirement put little pressure on bidders to make new bids once there were bids on most licenses. Bidding almost ended after 10 rounds but dramatically increased after the FCC raised the activity requirement in round 12. The elegance and the coherence of the proposal were not sufficient to make it an easy sell at the FCC. Many staff had little taste for taking the chance on an auction design that had never been used and seemed far more complex than any auction they had heard of. Chairman Reed Hundt’s legal advisor, Diane Cornell, argued that the mechanism, espe- cially the activity rule, was much too difficult for bidders to understand. I promised her that we would develop bidding software that would au- tomatically calculate activity requirements and make it easy for bidders to participate. At the time, no such software existed, but fortunately we were able to develop user friendly interfaces in time for the first auction. A more serious concern was that the auction might be an operational fiasco. If that happened, the argument that the design had theoretical beauty would not carry much weight in a congressional oversight hear- ing. My boss was quite frank when he told me that he did not want the FCCtobea“beta test site” for new auction designs. Why did the FCC adopt the basic Milgrom–Wilson auction design despite these concerns? First, it was good policy. It seemed to provide P1: FCH/FFX P2: FCH/FFX QC: FCH/FFX T1: FCH CB610B-FOR CB610-Milgrom-v3 October 10, 2003 11:9
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bidders sufficient information and flexibility to pursue backup strate- gies to promote a reasonably efficient assignment of licenses, without so much complexity that the FCC could not successfully implement it and bidders could not understand it. Just having a good idea, though, is not enough. Good ideas need good advocates if they are to be adopted. No advocate was more persuasive than Paul Milgrom. He was so per- suasive because of his vision, clarity and economy of expression, ability to understand and address FCC needs, integrity, and passion for getting things right. He was able to translate his theoretical vision into coherent practical proposals and explain in plain English how all the pieces fit to- gether. He took the time to learn relevant institutional facts and to listen. He was willing and able to modify his proposals to address FCC con- cerns about auction length and destructive strategic behavior. He never used hard sell or oversold his results, and thus he engendered the trust of FCC staff. He was always responsive to the frenetic time pressures under which the FCC often operates – willing to talk about auction rules while he was on vacation, take desperate calls late at night, and visit the FCC on very short notice during that first year it was developing its auction design. As persuasive as Milgrom was, the FCC might not have been willing to risk adopting such a novel auction design without additional outside supporters. One was John McMillan, whom the FCC hired as a consultant to provide independent analysis of alternative auction designs. His re- port to the FCC (a revised version published in the Journal of Economic Perspectives in 1994) provided strong support for the Milgrom–Wilson design. And his calm manner and articulate explanations were reassur- ing to FCC staff that we were going in the right direction. Another ally was Preston McAfee, who helped solidify support for the Milgrom–Wilson design when he said that he preferred it to the simpler simultaneous design he had developed at a time when he underesti- mated the FCC’s ability to implement anything but the simplest auction design. More important was his suggestion to modify the Milgrom– Wilson proposal to permit bid withdrawals subject to a penalty. In a conference organized by Barry Nalebuff in January 1994 to help the FCC sort out alternative auction designs, McAfee proposed a simple way to reduce the exposure risk faced by bidders for licenses with strong com- plementarities. To discourage strategic insincere bidding, the Milgrom– Wilson design had not allowed for any bid withdrawals. However, when P1: FCH/FFX P2: FCH/FFX QC: FCH/FFX T1: FCH CB610B-FOR CB610-Milgrom-v3 October 10, 2003 11:9
xx Foreword
a collection of licenses is worth more than the sum of the licenses indi- vidually, bidders face the risk of paying too much for part of a package of licenses when the rest of the package is won by other bidders. The National Telecommunications and Information Administration (NTIA), whose role includes advising the White House on telecommunications policy, had proposed combinatorial auction mechanism to address this concern. The design, based on the work of Banks, Ledyard, and Porter (1989) and developed in a NTIA staff paper by Mark Bykowsky and Robert Cull, seemed far too complex for the FCC to implement in the time avail- able. As an alternative, McAfee proposed permitting bid withdrawals subject to a payment equal to the difference between the withdrawn bid and the subsequent high bid. Though the FCC did not adopt the NTIA proposal, the fact that the NTIA proposed a simultaneous auction design was helpful in building support for the Milgrom–Wilson design. It made that mechanism look like a reasonable middle ground between sequential ascending bid auc- tions and simultaneous ascending auctions with package bidding. In addition to their written comments, in January 1994, the NTIA jointly sponsored with Caltech a PCS auction design conference that brought FCC staff together with academic experimentalists as well as game theo- rists. Proposed and organized by Mark Bykowsky and John Ledyard, the conference provided additional support for the use of a simultaneous auction mechanism. The demonstration by David Porter of the combi- natorial auction mechanism proposed by NTIA helped show the feasi- bility of some form of electronic simultaneous auction. Perhaps most important was a presentation by Charles Plott of experimental evidence on the relative performance of sequential, simultaneous, and combina- torial auction designs. This research sponsored by PacTel at Paul Mil- grom’s suggestion, offered experimental evidence that when there were strong synergies among items, simultaneous auctions were better than sequential auctions, and combinatorial bidding was even better. Based on both the theory and experimental evidence, Ledyard persuasively argued that though it would be nice if the FCC implemented the combi- natorial mechanism he had helped design, the FCC could achieve most of the benefits with a simpler simultaneous design along the lines pro- posed by Milgrom and Wilson. Part of the explanation for the successful collaboration between out- side economists and the FCC in designing spectrum auctions was that P1: FCH/FFX P2: FCH/FFX QC: FCH/FFX T1: FCH CB610B-FOR CB610-Milgrom-v3 October 10, 2003 11:9
Foreword xxi
the initial responsibility for a design was given to the FCC’sOffice of Plans and Policy (OPP), which has a tradition of applying economics to public policy and tends to be far more open to new approaches than the operating bureaus. The OPP had been advocating the use of auctions for more than 10 years prior to the passage of the auction legislation, and was a logical home for a small team drawn from throughout the agency. One of the pillars of that team was Karen Wrege, an auction project manager, whom the FCC recruited from the Resolution Trust Corpora- tion. In 1993, it was not enough to convince FCC Chairman Reed Hundt that simultaneous multiple round auction was the best auction design. He had to be convinced that the FCC could implement it with the year mandated by Congress. Karen was able to visualize how the auction might work, convince Don Gips on Hundt’s staff that it could work, and – as part of a remarkable FCC team – make it work. Jerry Vaughan led the team with indomitable courage through many harrowing moments, such as a complete system failure the night before the start of FCC auc- tion #3. The team was too large for me to mention here all who de- serve credit, but some who deserve particular mention for making the Milgrom–Wilson auction design proposal a reality are the lawyers Kent Nakamura, Jonathan Cohen, and Jackie Chorney, the information tech- nology specialist John Giuli, the contracting officer Mark Oakey, and the economist Greg Rosston. Much credit for implementing the FCC auctions goes to the contrac- tors and consultants. Most of the programming for the electronic auction system was performed by outside contractors. After the first auction, the FCC hired a second economic theorist, Peter Cramton, to provide advice on refining the auction design and to develop a tool to help bidders and the FCC track the progress of the auction. We also contracted with a team of experimental economists from Caltech: Charlie Plott, John Ledyard, and Dave Porter. Without the help of Plott and Antonio Rangel, a first year graduate student, the contractor for the FCC’s first auction might not have succeeded in translating the FCC auction rules into software code. Caltech also tested the software used in the first and second FCC narrowband PCS auctions. As part of their “torture testing” they paid experiment participants a bonus for any error they could find in the software. Caltech also developed a clever method for manually checking all the calculations during the first FCC auction. Run by Rangel in parallel with the electronic auction system, this also provided a manual backup P1: FCH/FFX P2: FCH/FFX QC: FCH/FFX T1: FCH CB610B-FOR CB610-Milgrom-v3 October 10, 2003 11:9
xxii Foreword
that could have been put into service if the electronic system had failed. Fortunately it did not. The first FCC simultaneous multiple round auction began on July 25, 1994 in the Blue Room of the Omni Shoreham Hotel in Washington, DC. Bidding was conducted electronically on site. Despite the testing of the software, there was some trepidation about whether it would work. There was particular concern about the software for stage II of the activity rule. Thechiefprogrammerforthecontractorthatdevelopedthesoftwareand would run it during the auction said, in essence, “I am completely con- fident that the software will work properly in stage II, but do not try it.” We never found out, because the auction closed successfully in stage I. Every round, the FCC decided on how to set the bid increments on each license. We had a committee of three consultants to advise us: John McMillan, a theorist; Charlie Plott, an experimentalist; and Bill Steven- son, an auctioneer. We had five days to complete the auction before we would be kicked out of the ballroom so it could be used for a wedding. There was vigorous discussion about how large to make the bid incre- ments, how long to make the rounds, and whether to deploy stage II of the activity rule. As it turned out, with few licenses, vigorous compe- tition, and bidders on site, the auction closed after 47 rounds and five days, in time for the wedding in the Blue Room. Perhaps the biggest hero of the story of putting auction theory to work is FCC Chairman Reed Hundt. He defied the traditional tendency of government bureaucracies to do the safe thing even if it is not the best thing. He always wanted to know: “What does economic theory tell us?” He always tried to put into practice his favorite motto, “Do the right thing.” But without economic theorists like Paul Milgrom, he would not have known what that was.
Evan Kwerel January 2003 P1: FCH/FFX P2: FCH/FFX QC: FCH/FFX T1: FCH CB610B-FOR CB610-Milgrom-v3 October 10, 2003 11:9
PUTTING AUCTION THEORY TO WORK
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CHAPTER ONE Getting to Work
The era of putting auction theory to work began in 1993–1994, with the design and operation of the radio spectrum auctions in the United States. Although the economic theory of auctions had its beginnings in the 1960s, early research had little influence on practice. Since 1994, auction theorists have designed spectrum sales for countries on six continents,
electric power auctions in the United States and Europe, CO2 abatement auctions, timber auctions, and various asset auctions. By 1996, auction theory had become so influential that its founder, William Vickrey, was awarded a Nobel Prize in economic science. In 2000, the US National Sci- ence Foundation’s fiftieth anniversary celebration featured the success of the US spectrum auctions to justify its support for fundamental re- search in subjects like game theory. By the end of 2001, just seven years after the first of the large modern auctions, the theorists’ designs had powered worldwide sales totaling more than $100 billion. The early US spectrum auctions had evolved into a world standard, with their major features expressed in all the new designs. It would be hard to exaggerate how unlikely these developments seemed in 1993. Then, as now, the status of game theory within eco- nomics was a hotly debated topic. Auction theory, which generated its main predictions by treating auctions as games, had inherited the controversy. At the 1985 World Congress of the Econometric Society, a debate erupted between researchers studying bargaining, who were skeptical that game theory could explain much about bargaining or be useful for improving bargaining protocols, and those investigating in auctions and industrial organization, who believed that game theory was illuminating their studies. Although game theory gained increasing prominence throughout the 1980s and had begun to influence the
1 P1: FCH/FFX P2: FCH/FFX QC: FCH/FFX T1: FCH CB610B-01 CB610-Milgrom-v3 October 6, 2003 11:5
2 Getting to Work
leading graduate textbooks by the early 1990s, there was no consensus about its relevance in 1994, when the Federal Communications Com- mission conducted the first of the new spectrum auctions. Thetraditionalfoundationsofgametheoryincorporatestarkassump- tions about the rationality of the players and the accuracy of their ex- pectations, which are hard to reconcile with reality. Yet, based on both field data and laboratory data, the contributions of auction theory are hard to dispute. The qualitative predictions of auction theory have been strikingly successful in explaining patterns of bidding for oil and gas1 and have fared well in other empirical studies as well. Economic labora- tory tests of auction theory have uncovered many violations of the most detailed theories, but several key tendencies predicted by the theory find significant experimental support.2 Taken as a whole, these findings indicate that although existing theories need refinement, they capture important features of actual bidding. For real-world auction designers, the lesson is that theory can be helpful, but it needs to be supplemented by experiments to test the applicability of key propositions and by prac- tical judgments, seasoned by experience. Whatever the doubts in the academy about the imperfections of game theory, the dramatic case histories of the new auctions seized public attention. An article in 1995 in the New York Times hailed one of the first US spectrum auctions3 as “The Greatest Auction Ever.”4 The British spectrum auction of 2000, which raised about $34 billion, earned one of its academic designers5 a commendation from the Queen and the title “Commander of the British Empire.” In the same period, game theorists were plying their trade on another important application as well. The National Resident Matching Program, by which 20,000 US physicians are matched annually to hospital residency programs, implemented a new design in 1998 with the help of the economist–game theorist Alvin Roth. By the mid-nineties, thirty-five years of theoretical economic research about fine details of market design was suddenly bearing very practical fruits.
1 See Hendricks, Porter, and Wilson (1994). 2 See Kagel (1995). 3 The design was based on suggestions by Preston McAfee, Paul Milgrom, and Robert Wilson. 4 William Safire, “The Greatest Auction Ever,” New York Times, March 16, 1995, page A17, commenting on FCC auction #4. 5 The principal designers were Professors Ken Binmore and Paul Klemperer. They give their account of the auction in Binmore and Klemperer (2002). It was Binmore whom the Queen of England honored with a title. P1: FCH/FFX P2: FCH/FFX QC: FCH/FFX T1: FCH CB610B-01 CB610-Milgrom-v3 October 6, 2003 11:5
1.2 Designing for Multiple Goals 3
1.1 Politics Sets the Stage To most telecommunications industry commentators, the main signif- icance of the US spectrum auctions was that a market mechanism was usedatall.Spectrumrights(licenses)intheUnitedStatesandmanyother countries had long been assigned in comparative hearings, in which regulators compared proposals to decide which applicant would put the spectrum to its best use. The process was hardly objective: it involved lawyers and lobbyists arguing that their plans and clients were most de- serving of a valuable but free government license.6 With its formal proce- dures and appeals, a comparative hearing could take years to complete. By 1982, the need to allocate many licenses for cellular telephones in the US market had overwhelmed the regulatory apparatus, so Congress agreed to allow licenses to be assigned randomly among applicants by lottery. The lottery sped up the license approval process, but it created a new set of problems. Lottery winners were free to resell their licenses, encour- aging thousands of new applicants to apply for licenses and randomly rewarding many with prizes worth many millions of dollars. Lottery win- ners were often simple speculators with no experience in the telephone industry and no intention of operating a telephone business. Economic resources were wasted on a grand scale, both in processing hundreds of thousands of applications and in the consequent need for real wire- less operators to negotiate and buy licenses from these speculators. The lotteries of small licenses contributed to the geographic fragmentation of the cellular industry, delaying the introduction of nationwide mobile telephone services in the United States. A better process was needed, and in 1993, Congress authorized auc- tions as the answer. The question of how an auction market for radio spectrum should be designed was left to the Federal Communications Commission (FCC).
1.2 Designing for Multiple Goals Congress did provide some instructions to the FCC governing the new spectrum auctions. One was that the first auctions were to be begun by July 1994. A second called for the auctions to promote wide partici- pation in the new industry. The FCC initially responded to the second
6 The process was once characterized by an FCC Commissioner as “the FCC’s equivalent of the Medieval trial by ordeal” (as quoted by Kwerel and Felker (1985)). P1: FCH/FFX P2: FCH/FFX QC: FCH/FFX T1: FCH CB610B-01 CB610-Milgrom-v3 October 6, 2003 11:5
4 Getting to Work
mandate by introducing bidding credits and favorable financing terms for small businesses and woman- and minority-controlled businesses, to reduce the cost of any licenses acquired by those businesses. The statute also specified that the auction process should promote “efficient and in- tensive use” of the radio spectrum, in contrast with the fragmented use promoted by the lottery system. The meaning of the word “efficient” was initially subject to debate, but it was eventually read in economic terms to mean, in the words of Vice President Albert Gore, “putting licenses into the hands of those who value them the most.”7 There is a powerful tradition in economics claiming that individuals and firms, left to their own devices and operating in a sound legal frame- work, tend to implement efficient allocations. The argument is that when resources are allocated inefficiently, it is possible for the parties to get together to make everyone better off. So, following their mutual inter- ests, the parties will tend to eliminate inefficiencies whenever they can. This traditional argument has its greatest force when the parties can all see what is required and have no trouble negotiating how to divide the gains created by the agreement. For radio spectrum, with thousands of licenses and hundreds of participants involved, computing just one efficient allocation can be an inhumanly hard problem, and getting par- ticipants to reveal the information about their values necessary to do that computation is probably impossible. Compared to the development of a universal standard (GSM) for mobile telephones in Europe, the more fragmented system that emerged in the United States highlights that the lottery system did not lead to efficient spectrum allocations. With so many parties and interests involved, the market took many years to recover from the initial fragmentation of spectrum ownership. During those years, investments were delayed and consumer services degraded. Getting the allocation right the first time does matter. Achieving that with an auction system called for a different and innovative approach. The FCC, which the law had charged with designing and running the spectrum auctions, had no previous auction experience. Within the FCC, the design task was assigned to a group led by Dr. Evan Kwerel, an economist and long-time advocate of using auctions to allocate spec- trum licenses.8
7 Quoted from Vice President Gore’s speech at the beginning of FCC auction #4. 8 Kwerel’s initial advocacy is explained in Kwerel and Felker (1985). P1: FCH/FFX P2: FCH/FFX QC: FCH/FFX T1: FCH CB610B-01 CB610-Milgrom-v3 October 6, 2003 11:5
1.2 Designing for Multiple Goals 5
Like any other important FCC decision, the auction design decision would need to be based on an adequate public record – a requirement that would force the FCC to go through a long series of steps. It would need to write and issue a proposed rule, allow a period for Comments and another for Reply Comments, meet with interested parties to dis- cuss and clarify the points of disagreement, resolve those disagreements, issue a ruling, consider appeals, and finally run the auction. Steps like these often stifle innovation, but that is not what happened on this oc- casion. With no political guidance about what kind of auction to use, no in-house experts lobbying to do things their way, and no telecom with an historically fixed position about how an auction should be run, Dr. Kwerel had unusual freedom to evaluate a wide range of alter- natives. Kwereldraftedanoticethatproposedacomplexauctionrule.Industry participants, stunned by the novel proposal and with little experience or expertise of their own, sought the advice of academic consultants. These consultants generated a flood of suggestions, and the FCC hired its own academic expert, John McMillan, to help them evaluate the proposed designs. In the end, Kwerel favored a kind of simultaneous ascending auction, based in large part on a proposal by Robert Wilson and me and a similar proposal by Preston McAfee. The Milgrom–Wilson–McAfee rules called for a simultaneous multiple round ascending auction.9 This is an auction for multiple items in which bidding occurs in a series of rounds. In each round, bidders make sealed bids for as many spectrum licenses as they wish to buy. At the end of each round the standing high bid for each license is posted along with the minimum bids for the next round, which are computed by adding a pre-determined bid increment, such as 5% or 10%, to the standing high bids. These standing high bids remain in place until superseded or withdrawn.10 An activity rule limited a bidder’s ability to increase its activity late in the auction, thus providing anincentivetobidactivelyearlyintheauction.Forexample,abidderthat
9 The principal difference was that the Milgrom–Wilson design proposed the now standard features that bidding on all licenses would remain open until the end of the auction, with progress ensured by Milgrom’s activity rule. McAfee’s design had no activity rule, and en- sured the progress of the auction by closing bidding on each license separately after a period with no new bids on that license. 10 A bidder that withdraws its bid pays a penalty equal to the difference, if positive, between the eventual sale price for the license and the amount of its withdrawn bid. If the eventual price exceeds its bid, then no penalty is payable. P1: FCH/FFX P2: FCH/FFX QC: FCH/FFX T1: FCH CB610B-01 CB610-Milgrom-v3 October 6, 2003 11:5
6 Getting to Work
has been actively bidding for ten licenses may not, late in the auction, begin bidding for eleven licenses. The theory of simultaneous ascending auctions is best developed for the case when the licenses being sold are substitutes. During the course of the auction, as prices rise, bidders who are outbid can switch their demands to bid for cheaper licenses, allowing effective arbitrage among substitute licenses. One of the clearest empirical characteristics of these auctions is that licenses that are close substitutes sell for prices that are also close – a property that is not shared by most older auction designs. The initial reception to Kwerel’s recommendation was skeptical. The proposed auction was unexpectedly complicated, and FCC Chairman Reed Hundt sought the advice of other FCC staff. He asked the economics staff: If you could pick any design you want, would this be it? He asked those who would have to run it: Can this really work? Even in the short time available to set it up? With the endorsement of his staff, Chairman Hundt decided to take the risk of adopting a new auction design.
1.2.1 Substitutes and Complements Auctions are processes for allocating goods among bidders, so the chal- lenge of auction design can only be understood by studying the demands of the participants. In the initial PCS auction, there were three groups of potential bidders. The first group included long-distance companies with no existing wireless businesses. These companies, including MCI and Sprint, were making plans to enter the wireless business on a na- tional scale. Each wished to acquire a license or licenses that would cover the entire United States, allowing it to make its service ubiquitous and to combine wireless with its own long distance service to offer an attractive and profitable package to consumers. A second group comprised the existing wireless companies, including AT&T, some regional Bell operating companies, and others. The compa- nies in this group already owned or controlled licenses that enabled them to offer services to parts of the country. Their objectives in the auction were to acquire licenses that filled in the varying gaps in their existing coverage and to expand to new regions or perhaps the entire na- tion. These companies posed a regulatory challenge for the FCC, which wanted to allow them to meet their legitimate business needs with- out gaining control of enough of the spectrum to manipulate market prices. To avoid this outcome, the FCC imposed limits on the amount of P1: FCH/FFX P2: FCH/FFX QC: FCH/FFX T1: FCH CB610B-01 CB610-Milgrom-v3 October 6, 2003 11:5
1.2 Designing for Multiple Goals 7
spectrum that any company could control in any geographic area. These existing wireless operators would be ineligible to bid for a nationwide PCS license of the sort that had typically been awarded in European countries. From MCI’s perspective, this meant that a nationwide license might be bought cheaply at auction, so it lobbied the FCC to structure the new licenses in this way. The last group consisted mainly of new entrants without wireless businesses. Some of these companies, like Pacific Bell in California, were quite large. These companies typically sought licenses or pack- ages covering large regional markets, but not licenses covering the entire nation. One of the first lessons to take from this description is that the auc- tion game begins long before the auction itself. The scope and terms of spectrum licenses can be even more important than the auction rules for determining the allocation, because a license can directly serve the needs of some potential bidders while being useless to others. For the actual PCS auctions, a license provided its owner the right to transmit and receive radio signals suitable for mobile telephone service in a par- ticular band of radio frequencies and in a particular geographic area. These license specifications constrained the possible spectrum alloca- tions. For example, suppose three separate licenses covering areas A, B, and C were put up for sale. If one bidder wanted a license covering A and half of B while the other wanted a license covering C and the other half of B, the license specifications would prevent each bidder from ac- quiring its optimal allocation. One task of the auction designer was to promote the best (most “efficient”) possible allocation, subject to such constraints. Achieving efficiency involves various subtle complications. A certain license may be valuable to one bidder because it helps exclude entry and increase monopoly power, but be valuable to another because the buyer will use it to create valuable services. In comparing the efficiency of allocations, only the second kind of value counts, but bidders do not respect that difference when placing their bids. The value of a license to a bidder may depend not only on the license itself, but also on the identities of other licensees and the technologies they use. For exam- ple, the licensee identities can affect their “roaming arrangements”– which allow their customers to use another company’s services when they roam to the other company’s license area. A third complication is P1: FCH/FFX P2: FCH/FFX QC: FCH/FFX T1: FCH CB610B-01 CB610-Milgrom-v3 October 6, 2003 11:5
8 Getting to Work
that the bidders may need to pool information even to determine their own likely profits from various arrangements, for example because the bidders have different information about the available technology or forecasted demand. But the fundamental barrier to efficiency that was most debated among the FCC auction designers concerned the packaging problem. The value of a license to a bidder is not fixed; it generally depends on the other licenses the bidder receives. For example, a bidder might be willing to pay much more per license for a package of, say, five or six licenses than for smaller or larger packages.11 Until such a bidder knows all of the licenses it will have, it cannot say how much any particular license is worth. Consider a situation with just two licenses. If acquiring one license makes a bidder willing to pay less for the second, then the licenses are substitutes. If acquiring one makes the bidder willing to pay more for the second, then the licenses are complements. With more than two li- censes, there are other important possibilities, and these add consid- erable complexity to the real auction problem. For example, if there are three licenses – say A, B, and C – and a certain bidder anticipates needing exactly two of them to establish its business, then A and B are complements if the bidder has not acquired C, but they are substitutes if the bidder has already acquired C. Nevertheless, most economic discus- sions of the auction design are organized by emphasizing the two pure cases. Recent auctions devised by economic theorists differ from their pre- decessors in the ways they deal with the problems of substitutes and complements. Our later analyses will show that some of the new designs deal effectively with cases in which the items to be traded are substi- tutes, but that all auctions perform significantly worse when licenses might either be substitutes or complements. The impaired performance may take various forms including a loss of efficiency of the outcomes, uncompetitively low revenues to the seller, vulnerability to collusion, complexity for the bidders, and long times to completion.
11 An instance of this sort arose in the Netherlands spectrum auction in 1998, where most of the licenses were for small amounts of bandwidth. New entrants were expected to need five or six such licenses to achieve efficient scale and make entry worthwhile. P1: FCH/FFX P2: FCH/FFX QC: FCH/FFX T1: FCH CB610B-01 CB610-Milgrom-v3 October 6, 2003 11:5
1.2 Designing for Multiple Goals 9
To illustrate how value interdependencies affect proper auction de- sign, we turn to a case study in which the matter received too little attention.
1.2.2 New Zealand’s Rights Auction New Zealand conducted its first auctions of rights to use radio spectrum in 1990. Some of the rights took the traditional form of license rights to use the spectrum to provide a specific service, such as the right to broadcast television signals using those frequencies. Others consisted of management rights according to which the buyer may decide how to use the spectrum, choosing, for example, television broadcasts, wireless telephones, paging, or some other service. In theory, when management rights are sold, private interests have an incentive to allocate spectrum to its most profitable uses, but the problem of coordinating uses among licensees can also become more complex. Acting on the advice of a consulting firm – NERA – the New Zealand government adopted a second-price sealed-bid auction for its first four auction sales. As originally described by Vickrey (1961), the rules of the second-price auction are these: Each bidder submits a sealed bid. Then, the license is awarded to the highest bidder for a price equal to the second highest bid, or the reservation price if only one qualifying bid is made. The auction gets its name from the fact that the second highest bid determines the price. The idea of a second-price sealed-bid auction strikes many people as strange when first they hear about it, but on closer analysis, the auction is not strange at all. In fact, it implements a version of the ascending (English) auction12 similar to the one used at Amazon.13 In an ascending auction, if a bidder has a firm opinion about what the item is worth, then he can plan in advance how high to bid – an amount that we may call the bidder’s reservation value. At sites like eBay and
12 The most common form of an ascending (English) auction is one in which the auctioneer cries out increasing bids and the bidders drop out when they are no longer willing to pay above the current price. The auction ends when there is just one remaining bidder. As the winning bidder is required to pay the current high price, it is optimal for each bidder to stay in the auction only until the current price is equal to his valuation (“reservation value”)of the item and not thereafter. 13 eBay also runs a similar auction, but its fixed ending time involves additional gaming issues as described by Roth and Ockenfels (2000). P1: FCH/FFX P2: FCH/FFX QC: FCH/FFX T1: FCH CB610B-01 CB610-Milgrom-v3 October 6, 2003 11:5
10 Getting to Work
Amazon, the bidder can instruct a proxy bidder to carry out a reservation value strategy. The proxy keeps beating the current highest bid on the bidder’s behalf so long as that bid is less than the specified reservation value. If everyone bids that way, then the outcome will be that compe- tition ends when the price rises to the second highest reservation value, or thereabouts (with differences due to the minimum bid increment). If everyone adopts such a reservation value strategy, then the ascending auction is almost the same as a second-price auction. Strategic considerations in a second-price auction are easy: each bid- der should set his reservation value to what the object is worth to him. If it happens that the highest bid among the other bidders is greater than this value, then he cannot do better than to bid his reservation value, because there is no bid he could make that would win the auction prof- itably. If, instead, it happens that the highest competing bid is less than his value, then setting his reservation value in this way wins and fixes the price at what the competitor bid, which is the best outcome that any bid could achieve. Thus, regardless of the bids made by others, setting a reservation value equal to the bidder’s actual value always earns at least as much as any other bid. The second-price sealed-bid auction has two advantages over most other designs. First, it duplicates the outcome of an ascending bid auc- tion with small bid increments, but without requiring the bidders to be assembled together or to hire agents to represent them in their absence. Second, it presents each bidder with a simple strategic bidding prob- lem: each merely has to determine his reservation value and bid it. This also means that there is no need for any bidder to make estimates of the number of other bidders or their values, for those have no bearing on a rational bidder’s optimal bid. The second-price auction has a simple extension to sales of multiple identical items, and it, too, can be motivated by considering a particular ascending auction. For example, suppose there is such an auction rule withsevenidenticalitemsforsale,tobeawardedtothesevenhighestbid- ders in an ascending outcry auction. Again, bidders might sensibly adopt reservation value strategies, bidding just enough to be among the top sevenbiddersanddroppingoutwhentherequiredbidfinallyexceedsthe bidder’s value. An analysis much like the preceding one then leads to the conclusion that the items will be awarded to the seven bidders with the highest values for prices approximately equal to the eighth highest P1: FCH/FFX P2: FCH/FFX QC: FCH/FFX T1: FCH CB610B-01 CB610-Milgrom-v3 October 6, 2003 11:5
1.2 Designing for Multiple Goals 11
value. To duplicate that with a sealed-bid auction, the rule must award items at a uniform price equal to the highest rejected bid. In such an auc- tion, the right advice to bidders is simple: “Bid the highest price you are willing to pay.” A similar uniform-price rule has sometimes been used in the sale of U.S. Treasury bills.14 In New Zealand, the government was selling essentially identical li- censes to deliver television signals. On the advice of its consultants, it did not adopt this highest-rejected-bid rule, but chose instead to conduct simultaneous second-price sealed tender auctions for each license. New Zealand’s second-price rules would work well in one case only: when the values of the items were independent – neither substitutes nor comple- ments. In the actual New Zealand auction, it would have been difficult to give bidders good advice. Should a bidder bid for only one license? If so, which one? If everyone else plans to bid for just one license and picks randomly, perhaps there will be some license that attracts no bids. Bidding a small amount for every license might then be a good strategy. On the other hand, if many spread around small bids like that, then bid- ding a moderate amount for a single license would have a high chance of success. With licenses that are substitutes or complements, indepen- dent auctions inevitably involve guesswork by the bidders that interferes with an efficient allocation. The actual outcome of the first New Zealand auction is shown in Table1.Noticethatonebidder,SkyNetworkTV,consistentlybidandpaid much more for its licenses than other bidders. The Totalisator Agency Board, which bid NZ$401,000 for each of six licenses, acquired just one li- cense at a price of NZ$100,000, while BCL, which bid NZ$255,000 for just one license, paid NZ$200,000 for it. Without knowing the exact values of various numbers of licenses to the bidders, it is impossible to be cer- tain that the resulting license assignment is inefficient, but the outcome certainly confirms that the bidders could not guess one another’s behav- ior. If Sky Network, BCL, or United Christian had been able to guess the pattern of prices, they would have changed the licenses on which they had bid. The bid data shows little connection between the demands expressed by the bidders, the numbers of licenses they acquired, and the prices they eventually paid, suggesting that the outcome was inefficient.
14 The Treasury rule sets a uniform price equal to the lowest accepted bid. P1: FCH/FFX P2: FCH/FFX QC: FCH/FFX T1: FCH CB610B-01 CB610-Milgrom-v3 October 6, 2003 11:5
12 Getting to Work
Table 1. Winning Bidders on Nationwide UHF Lots: 8 MHz License Rights Lot Winning Bidder High Bid (NZ$) Second Bid (NZ$) 1 Sky Network TV 2,371,000 401,000 2 Sky Network TV 2,273,000 401,000 3 Sky Network TV 2,273,000 401,000 4 BCL 255,124 200,000 5 Sky Network TV 1,121,000 401,000 6 Totalisator Agency Board 401,000 100,000 7 United Christian Broadcast 685,200 401,000
Source: Hazlett (1998).
A second problem was even more embarrassing to New Zealand’s government officials.15 McMillan (1994) described it as follows: “In one extreme case, a firm that bid NZ$100,000 paid the second-highest bid of NZ$6. In another the high bid was NZ$7 million and the second bid was NZ$5,000.” Total revenue, which consultants had projected to be NZ$250 million, was actually just NZ$36 million. The second- price rules allowed public observers to get a good estimate of the win- ning bidders’ profits, some of which were many times higher than the price. To avoid further embarrassment, the government shifted from the second-price sealed-bid format to a more standard first-price sealed- bid format, in which the highest bidder pays the amount of its own bid. As we will see later in this book, that did not guarantee higher prices. It did, however, conceal the bidders’ profits from a curious public. The change in auction format still failed to address the most serious auction design problems. Unlinked auctions with several licenses for sale that may be substitutes or complements force a choice between the risks of acquiring too many licenses and of acquiring too few, leaving a guessing game for bidders and a big role for luck. Allocations are unnec- essarily random, causing licenses to be too rarely assigned to the bidders who value them the most.
15 For a detailed account, see Mueller (1993). P1: FCH/FFX P2: FCH/FFX QC: FCH/FFX T1: FCH CB610B-01 CB610-Milgrom-v3 October 6, 2003 11:5
1.2 Designing for Multiple Goals 13
1.2.3 Better Auction Designs In the New Zealand case, alternative auction designs could have per- formed much better. For example, the government could have mimicked the design of the Dutch flower auctions. The winner at the first round would be allowed to take as many lots as it wished at the winning price. Once that was done, the right to choose next could be sold in the next auction round, and so on. No bidder would be forced to guess about which licenses to bid on with such an auction. Each bidder could be sure that, if it wins at all, it will win the number of lots or licenses anticipated by its business plan at the bid price it chose. There are other designs, as well, that limit the guesswork that bidders face. A common one in US on-line auctions allows bidders to specify both a price and a desired quantity. The highest bidders (or, in case of ties, those who bid earliest) get their orders filled in full, with only the last winning bidder running the risk of having to settle for a partial order. As with the Dutch design, efficiency is enhanced because bidders do not have to ponder over which licenses to bid on, and such rules reduce the exposure risk that a bidder may wind up acquiring licenses at a loss, because it buys too few to build an efficiently scaled system.
1.2.4 The FCC Design and Its Progeny In the circumstances of the FCC’s big PCS auction, it was obvious that some licenses would be substitutes. For example, there would be two licenses available to provide PCS service to the San Francisco area. Be- cause the two licenses had nearly identical technical characteristics and because, for antitrust reasons, no bidder would be allowed to acquire both, these licenses were necessarily substitutes. The argument that some licenses were complements was also made occasionally, but the force of the argument was reduced by the large geographic scope of the licenses.16 As in the New Zealand case, the main design issue was to minimize guesswork, allowing bidders to choose among substitute licenses based
16 Dr. Mark Bykowsky of the National Telecommunications and Information Administration (NTIA) was a forceful advocate of the view that licenses could be complements and pro- posed a complex package auction design to accommodate the possibility. His case that complementarity was important is more convincing for the later auctions in which smaller licenses were sold. Nonetheless, the short time available to run the first auction led to a near-consensus that the package auction proposal involved too many unspecified details and unresolved uncertainties for it to be evaluated and adopted immediately. P1: FCH/FFX P2: FCH/FFX QC: FCH/FFX T1: FCH CB610B-01 CB610-Milgrom-v3 October 6, 2003 11:5
14 Getting to Work
on their relative prices. When substitute goods are sold in sequence, either by sealed bids or in an ascending auction, a firm bidding for the firstitemmustguesswhatpriceitwillhavetopaylaterifitwaitstobuythe second, third, or fourth item instead. Incorrect guesses can allow bidders with relatively low values to win the first items, leading to an inefficient allocation. With this problem in mind, the final rules provided that the licenses would be sold all at once, in a single open ascending auction, during which bidders could place bids on any of the licenses and track bids on all the licenses. The openness of the process would eliminate the guesswork, allowing bidders to switch among substitute licenses, and promote equal prices for licenses that are perfect substitutes. In order for the auction to work in this idealized way, bidding on all licenses would need to remain open until no new bids were received for any license, but that raised a new issue. In a worst case scenario, the auction might drag on interminably as each bidder bid on just one license at a time, even when it was actually interested in eventually buying, say, 100 licenses. To mitigate this risk, the FCC adopted my activity rule. The general application of an activity rule involves two key concepts: eligibility and activity. A bidder’s activity in any round is the quantity of licenses on which it has either placed new bids in the round or had the high bid at the beginning of the round. In the example cited earlier, the quantity is just the number of licenses on which a bid is placed, but other quantity measures, including the total bandwidth of the licenses bid or the bandwidth multiplied by the population covered, have also been used. The rule specifies that a bidder’s total activity in a round can never exceed its eligibility. A bidder’s initial eligibility, applicable to the first round of the auction, is established by filing an application and paying a deposit before the bidding begins. Its eligibility in each later round depends on its recent bidding activity. One simple form of the rule specifies that a bidder’s eligibility in any round after the first is equal to its activity in the preceding round. Thus, bidders that are not active early in the auction lose eligibility to place bids later in the auction. This rule speeds the auction and helps bidders to make reliable inferences about the remaining demand at the current prices. The FCC rules have evolved since the original 1994 design, but larger changes have been made to adapt the simultaneous ascending auction to other applications. One common variation arises when there are many P1: FCH/FFX P2: FCH/FFX QC: FCH/FFX T1: FCH CB610B-01 CB610-Milgrom-v3 October 6, 2003 11:5
1.2 Designing for Multiple Goals 15
unitsofeachkindofitem,suchasauctionsinvolvingthesaleofelectricity contracts. In these auctions, for each item, each bidder bids its quantity demanded at the current price indicated on a clock visible to all bidders. The clock starts at a low price and keeps raising the price at any point at which the current total demand of all bidders exceeds the supply of that item. When demand equals supply on all items, the auction ends. A series of such clocks record the current prices for the various goods, and the rate of movement in these clocks determines the progress of the auction. A similar clock auction was used in March 2002 by the British
government to buy 4 million metric tons of CO2 emission reductions proposed by British businesses. Clock auctions share several key characteristics with their FCC an- cestor. Bidding on all items takes place simultaneously, so bidders can respond to changing relative prices. Prices rise monotonically, ensuring that the auction progresses in an orderly and predictable way. All bids are serious and represent real commitments. There is an activity rule that prevents a buyer from increasing its overall demand on all items as prices increase. Finally, bidding ends simultaneously on all the lots, so that opportunities for substitution do not disappear during the auction until all final prices are set. New variations based on the same principles continue to be created to solve a wide range of economic problems. Electricite´ de France (EDF) used a particularly interesting one in 2001 in a sale of electrical power contracts. The sale involved power contracts of different lengths, rang- ing from three months to two years, but all beginning at the same time – January 2002 for the first sale. Because different buyers wanted different mixesofcontractlengthsandbecauseallcontractscoveredthefirstquar- ter of 2002, EDF regarded the different kinds of contracts as substitutes. Lawrence Ausubel and Peter Cramton developed the auction design. ThefirststepwastoassistEDFindevelopingastandardfor“scoring”bids on contracts of different lengths. Bids expressed a price per megawatt per month that the buyer would pay for the right to acquire power. For the initial auction, EDF specified that the price for a three month con- tract for base-load power would always be €2139 higher than the corre- sponding price for a six month contract. Similarly, price differences were specified between the three-month contract and contracts lasting ten, twelve, twenty-four or thirty-six months. During the auction itself, the P1: FCH/FFX P2: FCH/FFX QC: FCH/FFX T1: FCH CB610B-01 CB610-Milgrom-v3 October 6, 2003 11:5
16 Getting to Work
price clocks were controlled to maintain these price relationships, for ex- ample, the price of a three-month contract was at all times €2139 higher than the price of a six-month contract. Prices for contracts of all lengths continued to rise until the total remaining demand exhausted the to- tal power available for the initial three-month period.17 Such an auction creates competition among bidders for contracts of different lengths, in- creasing both efficiency and sales revenue compared to more traditional auction designs. Recently, the EDF auction has been further modified to include a “supply curve,” so that total quantity of power sold depends on the price level.
1.3 Comparing Seller Revenues The question most frequently asked of auction designers is: What kind of auction leads to the highest prices for the seller? The answer, of course, depends on the particular circumstances, but even the thrust of the an- swer surprises many people: There is no systematic advantage of either sealed bid over open bid auctions, or the reverse. A particular formal statement of this conclusion is known as the payoff equivalence theorem. It holds that for an important class of auctions and environments, the average revenues and the average payoffs of bidders are exactly the same for every auction in the class. Toillustrate the logic of the idea, suppose you are selling an item that is worth $10 to bidder A and $15 to bidder B. If you sell the item using an ascending bid auction with both bidders in attendance, then bidder A will stop bidding at a price close to $10 and B will acquire the item for that price. If you use sealed bids instead and sell the item to the highest bidder, then the outcome will depend on what the bidders know when they bid. If they know all the values, then in theory B will bid just enough to ensure that it wins – around $10 or $10.01 – and A will likely bid close to $10. If they behave that way, the price will be just the same as in the ascending auction. The argument in this simple form was first made by Joseph Bertrand (1883). Nearly a century later, William Vickrey observed, that a similar conclusion holds on average for a much wider class of auction rules and in a more realistic set of situations than the one described here. For
17 For example, in the sale of power beginning January 2002, when the total demand exceeded the power available for the first quarter of 2002, the auction ended. Any remaining unsold power for, say, the second quarter of 2002 was then included in subsequent sales. P1: FCH/FFX P2: FCH/FFX QC: FCH/FFX T1: FCH CB610B-01 CB610-Milgrom-v3 October 6, 2003 11:5
1.3 Comparing Seller Revenues 17
forecasting average revenue, it is irrelevant which auction is used, within a certain class of standard auction designs. Experienced auctioneers often contest this irrelevance conclusion. Those who advocate ascending auctions argue that they generate more excitement and more competition than sealed bids. After all, they claim, no bidder is willing to bid close to its value unless pushed to do so by the open competition of the ascending auction design. Those who favor sealed bids counter by arguing that ascending auctions never result in more being paid than is absolutely necessary to win the auction; there is no money “left on the table.” Sealed bids frequently result in lots of money left on the table. For instance, in the December 1997 auction for licenses to provide wireless telephone services in Brazil, an international consortium including Bellsouth and Splice do Brazil bid $2.45 billion in that auction to win the license covering the Sao Paulo concession. This bid was about 60% higher than the second highest bid, so 40%, or about $1 billion, was left on the table.18 Similar arguments among practitioners arise quite frequently, some- times with variations. In the United States, the staff of the Treasury De- partment have periodically argued the relative merits of two alternative auction schemes for selling bills. In one scheme, each bidder pays the amount of its own bid for each bill it buys; in the other, all bidders pay the same market-clearing price, identified by the lowest accepted bid. Advo- catesofthefirst(“eachpaysitsownbid”)schemesaythatthegovernment will get more money from the auction, because winning bidders are by definition people who have bid more than the lowest acceptable bid. Advocates of the second (“uniform price”) scheme counter that bidders who know they must pay their own bid when they win will naturally bid less, reducing the market-clearing price and leading to lower revenues. Informal arguments like these show that the matter is subtle, but they do not settle the issue. A formal analysis based on the payoff equivalence theoremdiscussedinchapter3helpstocutthroughtheconfusion.Under certain idealized conditions, if the allocation of lots among bidders is the same for two different designs, then the average payoffs to all parties, including the average prices obtained by the seller, must also be exactly
18 Although the 60% overbid may be atypical, the ordinary amounts of money left on the table are still impressive. For example, in the Brazilian band A privatization, the median overbid was 27%. That is, for half the licenses, the winning bidders bid at least 27% more than the second highest bid. P1: FCH/FFX P2: FCH/FFX QC: FCH/FFX T1: FCH CB610B-01 CB610-Milgrom-v3 October 6, 2003 11:5
18 Getting to Work
the same. One cannot conduct a meaningful analysis of average prices alone, without also studying how the designs affect the distribution of the lots among the winning bidders. The practical uses of the payoff equivalence theorem are similar in kind to the uses of the Modigliani–Miller theorems in financial eco- nomics, the Coase theorem in contract theory, and the monetary neu- trality theorems in macroeconomics. All of these theorems assert that under idealized conditions, particular effects cannot follow from iden- tified causes.19 For example, according to the Modigliani–Miller theo- rems, if decisions about debt–equity ratios and dividend policies merely slice up the total returns to a firm’s owners without affecting the firm’s operations, then those decisions cannot affect the firm’s total market value. Today, financial economists explain financial decisions by focus- ing on how financial decisions might affect a firm’s operations – its taxes, bankruptcy costs, and managerial incentives. Similarly, according to the Coase theorem, if there were no costs or barriers to transacting, then the default ownership of an asset established by the legal system could not affect value. Today, economic theorists explain features of organization in terms of costs and barriers to transacting, including incomplete in- formation and incomplete contracts. The payoff equivalence theorem is similar: the payment terms of an auction do not affect the seller’s total revenue unless they are associated with a change in the allocation of the goods. Today, analysts focus more attention on how assumptions of the theorem are violated and the consequences of those violations or, for government regulators, about the implied trade-offs between their allocation and revenue objectives. The planning for a sale of electrical power in Texas in 2002 illustrates how the payoff equivalence theorem has been applied in practice. Ac- cording to the planned auction design, the auctioneer would gradually raise the prices for any products with excess demand and would ac- cept quantity demands from the bidders, in much the fashion that Leon Walras once described. The auctioneer would not tell the bidders the
19 According to the Modigliani–Miller theorems, under its idealized frictionless-markets con- ditions, a firm’s financial structure and dividend policy cannot affect its market value. Ac- cording to the Coase theorem, under other idealized conditions, the initial allocation of ownership rights cannot alter the efficiency of the final allocation. Monetary neutrality the- orems hold that under yet other idealized conditions, monetary policy cannot change real outcomes in an economy. The payoff equivalence theorem holds that under its idealized conditions, changing payment rules cannot affect the participants’final payoffs. P1: FCH/FFX P2: FCH/FFX QC: FCH/FFX T1: FCH CB610B-01 CB610-Milgrom-v3 October 6, 2003 11:5
1.4 The Academic Critics 19
quantities demanded by others. The rules called for the auctioneer to stop raising the price for a product when its total demand falls to the level of available supply. Texas ratepayers benefit from the revenues of this power sale, and the ratepayers’ advocate argued that the auctioneer should continue to raise prices until demand is actually less than supply, and should then roll back the price by one increment. The idea was to sell power for the highest market clearing price, rather than the lowest one. This rule was problematic for a variety of reasons relating to the details of the auction, and the design team cited the payoff equivalence theorem to argue that there was little reason to expect that the proposed change would lead to higher prices on average, because bidders would bid differently if the payment rules were changed. A bidder that knows it may acquire power at a lower price if it withdraws demand early will be more inclined to do so than a bidder that knows it cannot cause a price rollback. The net effect on revenues is hard to predict, because it depends on how the proposed new rule changes the allocation. Even- tually, the ratepayer advocate agreed not to oppose the auction design.
1.4 The Academic Critics Economists who are putting auction theory to work encounter a dazzling array of issues, from ideological to theoretical to practical. Recognizing the complexity of the problems and the short times available to solve them, the engineering work for auctions sometimes entails guesses and judgments that cannot be fully grounded in a complete economic analy- sis. Auction designers generate ideas using theory, test those ideas when they can, and implement them with awareness of their limitations, sup- plementing the economic analysis with worst case analyses and other similar exercises. The idea that economic theorists can add value through this mixture of auction theory and practical judgment has come under attack from some members of the economics profession. Some of the more frequent attacks, and my responses to them, are expressed below.
1.4.1 Resale and the Coase Theorem One of the most frequent and misguided criticisms of modern auction design comes in the form of the remarkable claim that the auction design does not matter at all. After all, say the critics, once the licenses are P1: FCH/FFX P2: FCH/FFX QC: FCH/FFX T1: FCH CB610B-01 CB610-Milgrom-v3 October 6, 2003 11:5
20 Getting to Work
issued, parties will naturally buy, sell, and swap them to correct any inefficiencies in the initial allocation. Regardless of how license rights are distributed initially, the final allocation of rights will take care of itself. Some critics have gone even farther, arguing on this basis that the only proper objective of the government is to raise as much money as possible in the sale, because it should not and cannot control the final allocation. Tojustify this argument, the critics relied on the Coase theorem, which holds that if there are no costs to transacting after the auction, then the initial allocation of property rights cannot affect the final alloca- tion, which will necessarily be efficient. Coase reasoned that so long as the allocation remains inefficient, the parties will continually find it in their interests to buy, sell, and swap as necessary to eliminate the inefficiency.20 The “zero transaction cost” assumption on which the Coasian argu- mentisbased,however,isnotonethatCoaseeveradvocatedasadescrip- tion of reality. Rather, it was advanced as part of a thought experiment to emphasize the importance of understanding actual transaction costs. Assuming that actual transactions costs are zero when they are not can lead to serious errors in one’s conclusions. The history of the US wire- less telephone service offers direct evidence that the fragmented and inefficient initial distribution of rights was not quickly correctable by market transactions. Despite demands from consumers for nationwide networks and the demonstrated successes of similarly wide networks in Europe, such networks were slow to develop in the United States. As I argued during the deliberations at the FCC, the conclusion that initial allocations do matter follows by juxtaposing two well-known propositions from economic theory.21 The first is that, as explained in chapter 2, auction mechanisms exist that achieve efficient license allo- cations for any number of available licenses, provided the government
20 The Coase theorem includes a variety of assumptions that may fail in this application, such as the assumption that the parties’ values reflect social value, not market power; the assumption that the parties have unlimited budgets, so spending on spectrum rights does not impair the ability to invest in infrastructure; and the assumption that rights have no externalities, that is, that bidders do not care about which competitors get license rights. The importance of the last assumption is analyzed by Jehiel and Moldovanu (1999). 21 The theory described here applies to private values models, in which a bidder’s maximum willingness to pay for any good or package of goods is independent of what other bidders know about that good. P1: FCH/FFX P2: FCH/FFX QC: FCH/FFX T1: FCH CB610B-01 CB610-Milgrom-v3 October 6, 2003 11:5
1.4 The Academic Critics 21
uses the auction from the start. With just one good for sale, the English auction is such a mechanism. The generalized Vickrey auction, which works even in the case of multiple goods, is introduced in chapter 2. The second proposition is that, even in the simplest case with just a single license for sale, there exists no mechanism that will reliably un- tangle an initial misallocation. Intuitively, in any two-sided negotiation between a buyer and seller, the seller has an incentive to exaggerate its value and the buyer has an incentive to pretend its value is lower. These misrepresentationscandelayorscuttleatrade.Accordingtoafamousre- sult in mechanism design theory – the Myerson–Satterthwaite theorem – there is no way to design a bargaining protocol that avoids this problem: delays or failures are inevitable in private bargaining if the good starts out in the wrong hands.
1.4.2 Mechanism Design Theory A second line of criticism emerges from a part of game theory called mechanism design theory.Amechanism is essentially a set of rules to govern the interactions of the parties. For example, it may specify the rules of an auction. Are there to be sealed or ascending bids? If sealed bids, how will the winner and price be determined? And so on. Once the rules of the mechanism and the designer’s objective have all been specified, the designer applies some criterion, or solution concept, to predict the outcome and then evaluates the outcome according to the objective. In the theory’s purest and most elegant form, the aim is to identify the mechanism that maximizes the performance according to the specified objective. For example, one might try to find the auction thatmaximizestheexpectedsellingpriceortheexpectedefficiencyofthe outcome. We will treat parts of this theory at length later in this book. Mechanism design theory poses this challenge to practical auction designers: how can you incorporate the use of theory without, at the same time, applying the mechanism design approach? If you believe the theory accurately describes the behavior of players, you should use it to optimize the mechanism performance. There is a longstanding joke about the arbitrage theory in financial economics that applies equally to mechanism design theory.Twopeople are walking along a street when one spots a $100 bill on the ground. “Pick it up,” says one. “Why bother?” replies the other. “If it were real, someone would have picked it up already!” P1: FCH/FFX P2: FCH/FFX QC: FCH/FFX T1: FCH CB610B-01 CB610-Milgrom-v3 October 6, 2003 11:5
22 Getting to Work
Like arbitrage theory, the equilibrium analysis of game theory is an abstraction based on a sensible idea. Just as arbitrage theory implies that people do not leave real $100 bills lying on the street, equilibrium theory says that players in a game do not overlook ways to increase their payoffs. Both theories are useful idealizations – not reasons to leave $100 bills ly- ing on the ground. Theories like these, based on ubiquitious awareness and thoroughly rational calculations, are obviously inexact models of real behavior, and one should be especially careful about applying them to choices that are complex and subtle, even when the players are so- phisticated and experienced. In real auctions, where some players are unsophisticated, inexperienced, or lacking the time and other resources to support effective decision-making, equilibrium theory is still less reliable. Despite their drawbacks, equilibrium models can be very valuable to real-world mechanism designers. Just as a mechanical engineer whose mathematical model assumes a frictionless surface treats those calcula- tions as inexact, an economic designer whose model assumes that the players adopt equilibrium strategies can treat the predictions as approx- imations. Just as the real-world mechanical engineer pays attention to factors that increase friction and builds in redundancy and safety mar- gins, the real-world mechanism designer pays attention to timing and bidder interfaces to make rational decisions easier, and plans to accom- modate worst case scenarios, in case bidders make mistakes or simply behave contrary to expectations. In the present state of the art, academic mechanism design theory relies on stark and exaggerated assumptions to reach theoretical con- clusions that can sometimes be fragile. Among these are the assump- tions (i) that bidders’ beliefs are well formed and describable in terms of probabilities, (ii) that any differences in bidder beliefs reflect differences in their information, (iii) that bidders not only maximize, but also cling confidently to the belief that all other bidders maximize as well. These assumptions are extreme, and they are typically compounded in practice by the use of additional simplifying assumptions. Mechanisms that are optimized to perform well when the assumptions are exactly true may still fail miserably in the much more frequent cases when the assump- tions are untrue. Useful real-life mechanisms need to be robust. Those that are too fragile should be discarded, whereas a robust mechanism P1: FCH/FFX P2: FCH/FFX QC: FCH/FFX T1: FCH CB610B-01 CB610-Milgrom-v3 October 6, 2003 11:5
1.4 The Academic Critics 23
can sometimes be confidently adopted even if, in the corresponding mechanism design model, it is not provably optimal.22 Besides the very demanding behavioral assumptions that charac- terize the theoretical mechanism design approach, the existing formal models of mechanism design theory capture and analyze only a small subset of the issues that a real auctioneer faces. Some of the important issues that are usually omitted from mechanism design models are listed below. Although none of these are incompatible with mechanism design theory in principle, accounting for all in a single optimization model is far beyond the reach of present practice. r What to sell? If a farmer dies, should the entire farm be sold as a unit? Or should some fields be sold to neighbors? The house and barn as a holiday and weekend home? How should the FCC cut up the radio spectrum? Should power suppliers be required to bundle regulation services, or should they be priced separately? r To whom and when? Marketing a sale is often the biggest factor in its success. Bidders may need to study the opportunity and line up partners, financing, regulatory approvals, and so on. Conditions may change: financing may be more easily available at one time than an- other; uncertainties about technology or demand may become partly resolved; etc. Bidders may actively try to discourage others from bid- ding, hoping to get a better price.23 Auctioneers may seek to screen bidders to encourage participation by those who are most qualified, or may subsidize some participants to increase competition. r How? For example, if the deal is complicated and needs to be indi- vidually tailored for each bidder, a seller might prefer to engage in a sequence of negotiations to economize on costs. If an auction is to be used, the right kind can depend, as we have already seen, on whether the items are substitutes or complements.
22 The view expressed here is a variation of the Wilson doctrine, which holds that practical mechanisms should be simple and designed without assuming that the designer has very precise knowledge about the economic environment in which the mechanism will operate. Here, we further emphasize that even given a very complete description of the economic environment, the behavior of bidders cannot be regarded as perfectly predictable. 23 On the eve of the FCC PCS spectrum auction #4, the author made a television appearance on behalf of Pacific Bell telephone, announcing a commitment to win the Los Angeles telephone license, and successfully discouraging most potential competitors from even trying to bid for that license. P1: FCH/FFX P2: FCH/FFX QC: FCH/FFX T1: FCH CB610B-01 CB610-Milgrom-v3 October 6, 2003 11:5
24 Getting to Work
r Interactions? Decisions about what to sell, to whom, when, and how are not independent ones. What to sell depends on what buyers want, which depends on who is bidding, which may depend on how and when the auction is conducted. r Mergers and Collusion? The European spectrum auctions of 2000, with their very high stakes, provided some interesting examples of before-the-auction actions to reduce competition. In Switzerland, last minute mergers among potential bidders resulted in only four bidders showing up for four spectrum licenses. The auction was post- poned, but the licenses were eventually sold for prices close to the government-set minimum. Similar problems of valuable spectrum attracting few bidders and resulting in prices near the minimum oc- curred in Germany, Italy, and Israel. r Resale? Most of the theory of mechanism design starts with a given set of bidders that keep whatever they buy. The possibility of resale not only affects auction strategy, it may also attract speculators that buy with the intention of reselling. Should the seller encourage specula- tors, because additional bidders create more competition in the auc- tion? Or should the seller discourage them, because value captured by speculators must come from someone else’s payoff – possibly the seller’s?
The mechanism design purist’s view,which holds that the only consis- tent approach is to develop theoretically “optimal” mechanisms, is not useful in practice. Even if we could incorporate all the features described above, our models of human behavior are not nearly accurate enough for use in optimization. Behavior is neither perfectly stable over time, nor the same across individuals, nor completely predictable for any single individual. Useful analyses must be cognizant of these realities. Despite these limitations, a large portion of this book focuses on mechanism design and related analyses. The theory is especially use- ful in practice for identifying issues and effects. Among the decisions that the theory can illuminate are ones about information policy (what information to reveal to bidders), how to structure split awards (in which a buyer running a procurement auction splits its business between two or more suppliers), how to create scoring rules (in which bids are eval- uated on dimensions besides price), and when and how to implement handicapping (in which the auctioneer treats bids unequally in order P1: FCH/FFX P2: FCH/FFX QC: FCH/FFX T1: FCH CB610B-01 CB610-Milgrom-v3 October 6, 2003 11:5
1.4 The Academic Critics 25
to encourage more effective competition, for example, promote small businesses or those run by women and minorities). The mechanism de- sign approach also helps answer important questions about when to use auctions at all. Purchasing managers sometimes pose this question by asking whether particular goods and services are “auctionable,” that is, whether the most effective procurement process is to run a formal bidding process.
1.4.3 Theory and Experiment In sharp contrast to mechanism design purists, some economic experi- menters raise an opposite objection: why should any attention be paid to auction theory at all, now that we have the capability to test alterna- tive auction designs in experimental economics laboratories? Theories sometimes fail badly. The rest of the time, they explain only some of the data. So why rely on theory at all? The possibility of experimental tests has, indeed, fundamentally shifted the way auctions can be designed. In the FCC auction design, successful tests conducted by Charles Plott in his laboratory at Caltech helped convince the FCC to adopt the theoretically motivated Milgrom– Wilson design. Working software demonstrating the design was another important element.24 Yet, the experiments to date have been very far from replicating the actual circumstances of high-value auctions. In practice, it is unlikely that anyone will ever test a range of actual proposals in a completely realistic setting. The amounts at stake in ex- periments are necessarily much smaller, and the preparation time for bidders will normally be much less. Because experimental settings differ so much from the auctions they simulate, the role of theory is indispens- able. Theory guides the design of experiments, suggests which parts of any experimental results might be generalized, and illuminates the economic principles at work, enabling further predictions and improve- ments upon the original design.
24 Working software demonstrating the feasibility of the new design was another important element. Implementation issues also played a huge role in the debate. The very possibility of running the computer implemented simultaneous auction raised the hackles of critics in 1994. To rebut the critics, my assistant, Zoran Crnja, programmed a flawless small-scale version of the software in a set of linked Excel spreadsheets. His software convinced the FCC that a reliable system could be created using our proposed rules even in the short time available. P1: FCH/FFX P2: FCH/FFX QC: FCH/FFX T1: FCH CB610B-01 CB610-Milgrom-v3 October 6, 2003 11:5
26 Getting to Work
The philosopher Alfred North Whitehead, when asked whether theory or facts was more important, answered famously: “theory about facts.” Indeed, theories that are incompatible with facts are useless, but there can be no experimental designs and, indeed, no reporting of experimen- tal results without a conceptualization of the issues. Theory will always play a key role in answering engineering questions, including questions about auction design.
1.4.4 Practical Concerns The final criticism is that, in the real world, the whole mechanism design approach is irrelevant for several reasons. First, the auction rules them- selves are subject to bargaining: there is no single mechanism designer. Second, the rules are rarely even a first-tier concern in setting up and running a complex auction. Several other issues are more important. One such issue is marketing: an auction cannot succeed without par- ticipants. This interacts with the first observation: bidders may simply refuse to participate in designs that they consider strange or unfair.25 This very observation, however, emphasizes that good design can be an affirmative way to attract more and better participants. There are many examples of auctions and other competitions that get poor results because the rules are rigged to favor particular bidders and so discourage others from participating. One is the earlier description of MCI’s attempts to rig the US spectrum auctions in its favor by making the “lot” a single national license. When different bidders want different kinds of lots, a package auction design, such as the ones often used in bankruptcy sales, may enable wider participation. Anotherexampleistheinitialpublicoffering(IPO)ofsharesinayoung company. In the past, the investment banks that organize the IPOs have often reserved shares in “hot” offerings for the bank’s biggest and best customers, and that discourages small investors from participating. Try- ing to buck this trend, the investment bank WR Hambrecht has intro- duced its Open IPO product, which is a uniform price auction in which large and small investors are all subject to the same auction rules. The company tries actively to attract small investors to increase demand for
25 My own experience designing a procurement auction system for Perfect Commerce, Inc., revealed the seriousness of this concern. Sellers do often refuse to participate in auctions that are not structured to their liking. P1: FCH/FFX P2: FCH/FFX QC: FCH/FFX T1: FCH CB610B-01 CB610-Milgrom-v3 October 6, 2003 11:5
1.4 The Academic Critics 27
shares and create an alternative to the existing auction system, although its success will also depend on attracting larger investors, too, and com- panies willing to experiment with a new system. A second important practical issue concerns the property rights being allocated. For example, if auctions are to be used to allocate takeoff and landing rights at a congested airport, then the rights themselves need to be carefully defined. What is to happen to a plane that is delayed for mechanical reasons and cannot depart in its assigned slot? What are the airline’s rights if weather delays decrease the capacity of the airport? No sophisticated auction rule can lead to a good outcome unless these prac- tical issues are resolved, and an auction system that fails to coordinate all the resources needed by the airlines – takeoff slots, landing slots, rights through en route choke points, gate access, and so on – cannot succeed regardless of how well rights are defined. Real problems require com- prehensive solutions, and the auction rules are a partial solution whose importance varies across applications. Another important practical detail for electronic auctions is the in- terface used by bidders. The original FCC auction software made it easy for bidders to make mistakes. On several occasions, bidders made what came to be called “fat finger bids.” For example, when trying to bid $1,000,000, a bidder might accidentally enter a bid of $10,000,000 – an error encouraged by the fact that the early interfaces could not accept commas in the bid field. The FCC’s solution for this problem, however, was one that consid- ered more than the ease of bidding. Under the FCC’s initial rules, bid- ders found it easy to communicate messages, including threats, with their bids in the auction. Suppose, for example, that bidder A wished to discourage competitor B from bidding on a particular license, say #147, in a particular auction. If B bid on that license, A might retaliate by raising the price of another license on which B had the current high bid of, say, $9,000,000 by bidding $10,000,147, where the last three digits send a none-too-subtle message about its motivations. Such bids were frequently observed in some of the early FCC auctions. Both the “fat finger” and the signaling problems were solved when the FCC changed the auction interface to require that a bidder select its bid from a short dropdown menu on its bidding screen. All bids on the menu used round numbers, being the minimum bid plus one or more increments. This system eliminated typos involving one or more extra P1: FCH/FFX P2: FCH/FFX QC: FCH/FFX T1: FCH CB610B-01 CB610-Milgrom-v3 October 6, 2003 11:5
28 Getting to Work
digits and simultaneously made it much harder for bidders to encode messages in their bids. Some critics respond to such anecdotes with the claim that although they do show that rules matter, they mainly show the dangers in elec- tronic auctions or auctions using novel rules. However, even familiar, low-tech auctions can perform badly on account of problematic rules. In 1998, the Cook County, Illinois, tax collector conducted a traditional oral outcry auction to sell the right to collect certain 1996 property taxes that were two years overdue. In that 1996 tax sale auction, a bid specified the penalty rate that the winning bidder could charge in addition to the taxes due, as compensation for its collection services. The auction was conducted in an ordinary meeting room, with the auctioneer sitting in the front. The auctioneer would read a property number, and the bidding instantly began, with the bidders shouting penalty amounts. The max- imum opening bid was 18%, and successively lower bids were shouted until a winning low bidder was determined. The trouble occurred when several bidders simultaneously opened with bids of the maximum amount. Under the Cook County rules for that year, in the event of such a tie, the auctioneer was to assign the properties to winning bidders essentially at random. A bidder tied with, say, five others at 18% then faced a simple choice. He could bid less than 18%, having roughly a one in six chance to win the auction at a much lower rate than 18%. Or he could sit quietly and enjoy a one in six chance to win at a rate of 18%. Most bidders chose to sit quietly, and about 80% of the properties sold at the maximum rate of 18%. Howcanwebesureitwasthefaultyrules,ratherthancollusionamong (more than a dozen) bidders, that accounted for this outcome? A few days after the auction began, the county auctioneer announced a change in the rules. In the future, a tie bid at 18% would result in withdrawal of the property from the auction. After the change, penalty rates quickly collapsed to a lower level, providing some initial evidence that the treat- ment of ties does matter. Immediately after the rule change, some bid- ders sought a court order to restrain the auctioneer from changing its rules during the auction. The court agreed and issued the order. After the order was issued and the original rules restored, the winning bids quickly returned to 18%. Understanding auction theory is helpful for more than just avoiding obviously bad designs. Well-designed auctions that link the allocation P1: FCH/FFX P2: FCH/FFX QC: FCH/FFX T1: FCH CB610B-01 CB610-Milgrom-v3 October 6, 2003 11:5
1.4 The Academic Critics 29
of related resources can perform much better than traditional auction sales. In the New Zealand case described earlier, if the novel second-price auction rules had been replaced with more traditional pay-as-bid rules, any simultaneous sealed-bid auction would still be prone to misalloca- tion, because bidders would still need to guess about which TV licenses to bid on. Computational experiments suggest that 25–50% of the value might have been lost simply because the allocation was so poorly coordi- nated. In similar circumstances, the current world standard for spectrum auctions, the simultaneous ascending auction, can theoretically lead to a more efficient outcome and higher revenues as well. The simultaneous ascending auction has limitations too, which can be particularly important when the items for sale are ones that different bidders prefer to package in different ways, or when there are compli- cated constraints on the collection of acceptable offers. In such cases, a package auction design can both attract a wider set of bidders and vastly increase the likelihood that the right packages emerge from the auction. The design of these auctions is, however, subject to many pitfalls, to which we return in part II of this book. There are many more examples of the importance of the detailed auction rules. One is from a Mexican sealed-bid auction for a road con- struction contract, in which the bidders were asked to submit a total bid and to divide the total bid into three sub-parts in case part of the project was delayed or canceled. Although each bidder was required to specify four numbers (a price for each part and a total price), the project was to be awarded based only on the total price. The winning bidder submitted a bid in which the “total” was less than the sum of the prices for the three parts. As matters transpired, the sum of the three was low enough to win, and the winner claimed that he had simply made an “arithmetic mistake” and that the price must, of course, be the sum of the three component parts. It seems more likely that this device was used to place two bids, allowing the bidder to withdraw the lower bid if the higher one was a winner. That could be a useful option in a competitive setting, but even more so if the bidders were colluding, because the low “total price” bid would prevent a deviator from cheating on the agreement and placing a lower than expected bid. Indeed, if the auctioneer intended to facilitate collusion in the bidding, this was quite an effective design! Another example of how the details matter is drawn from the German experience in a 1999 spectrum auction. In that auction, Mannesmann P1: FCH/FFX P2: FCH/FFX QC: FCH/FFX T1: FCH CB610B-01 CB610-Milgrom-v3 October 6, 2003 11:5
30 Getting to Work
and T-Mobil managed to divide the market between themselves without engaging in intense price competition. With ten licenses for sale and a 10% price increment, Mannesmann opened the bidding by jumping to pricesofDM20millionforfivelicensesandDM18.18millionfortheother five, effectively suggesting to T-Mobil that the ten licenses be divided five-and-five at a price of DM20 million. In the event, T-Mobil bid DM20 million for the five lower-priced licenses and that ended the auction. The facts that equal division was possible and that the bidder could make such jump bids are design elements that contributed to this outcome. The risk was predictable. Indeed, the danger that such rules posed had been previously been pointed out in a 1997 report commissioned by the US spectrum authorities.26 In the US electricity markets, ill-considered market rules frequently contributed to high prices by making it too easy for power suppliers to manipulate the system. In a famous example, energy traders at the Enron Corporation manipulated the California power market by scheduling transmissions on congested links that were far in excess of those Enron had actually planned. That led the California Power Exchange to try to mitigate the expected congestion by paying the company to reduce its transmissions, resulting in massive profits for the company. Only after repeated failures did these designs evolve to produce more reasonable results, yet all of these defects could have been anticipated by a simple game theoretic analysis of the market designs. The most careful statistical evidence of the importance of design comes not from auction markets per se but from the closely related matching markets, such as the ones by which most new US doctors are matched to hospital residency programs. Roth (1991) provides evidence that a particular characteristic of the matching rules – whether the rules lead to a stable match – is an important determinant of whether or- ganized markets succeed in attracting participants over a long period of years. A match is stable if no medical student strictly prefers to be matched to another program rather than the one he is currently matched with while that other program simultaneously strictly prefers that med- ical student to one of those with whom it is matched. The analogous criterion for auctions is that no group of participants should be able to do better by rejecting the auction outcome and making a side deal of
26 See Cramton, McMillan, Milgrom, Miller, Mitchell, Vincent, and Wilson (1997). P1: FCH/FFX P2: FCH/FFX QC: FCH/FFX T1: FCH CB610B-01 CB610-Milgrom-v3 October 6, 2003 11:5
1.5 Plan for This Book 31
their own. Auctions that do not have this theoretical property are likely to run into trouble in practice, as some participants try to void the auction outcome to reach a better deal among themselves. Successful auction programs need to be well designed in all impor- tant respects, of which auction rules are one. Applying the perspective of auction theory can be valuable in many ways. It can enable an auction- eer to avoid mistakes like those that marred the 1993 spectrum auction in New Zealand, the 1996 tax auction in Cook County, the California electrical power markets and the 1999 German spectrum auction. It can help the auctioneer to pursue multiple objectives, like promoting mi- nority participation, encouraging alternative suppliers, and enhancing competition among bidders with diverse advantages. Finally, rules can be designed to accommodate complicated preferences and constraints for the bidders and the auctioneer. We will see some examples of this later in this book.
1.5 Plan for This Book This book integrates two projects, which are presented in the two main parts. The first part gives a review of traditional auction theory and is based on courses that I have given over a period of years at Stanford, Jerusalem, Harvard, and MIT. Traditional auction theory is based largely on the theory of mechanism design and the chapter organization fol- lows certain principles of that theory. Much of the analysis is focused on auctions in which each buyer wants only a single object – a condition called singleton demand. My treatment of the material differs from other treatments in two ways. First, it emphasizes practical applications where possible and makes an effort to include the issues that are most important in prac- tice. Second, the treatment reflects my view that incentive theory is not best viewed as some entirely new part of economics; it is best viewed as an evolutionary development of traditional demand theory. Rather than treating it from its own specialized perspective that obfuscates connec- tions with older theories, I use general perspectives and techniques that not only unify the theories but also prove their worth by reducing the lengthy and difficult proofs of incentive theory to shorter, more intuitive ones. The second part of the book differs from the first in its questions and methods. The questions mainly concern the design of auctions P1: FCH/FFX P2: FCH/FFX QC: FCH/FFX T1: FCH CB610B-01 CB610-Milgrom-v3 October 6, 2003 11:5
32 Getting to Work
for environments in which there are multiple heterogeneous goods. These environments are fundamentally more complex than ones with singleton demand. One reason is that the number of possible allocations is exponentially larger, which leads to serious issues about the practical feasibility of auction algorithms and bidder strategies. For example, in an auction with five bidders and one item, there are only five theoretically possible allocations of the item, and each bidder bids over just a single item. However, in an auction with five bidders and five items, there are 55 = 3125 theoretically possible allocations. A second way in which singleton demand is special is that it eliminates much of the tension between promoting efficient allocations and ensuring competitive rev- enues for the seller. In the general case of part II, where multiple hetero- geneous goods are sold with complementarities among the items, that tension can be severe. For example, the Vickrey auction, noted for its ability to promote efficient outcomes, can lead to zero or low revenues in relevant examples. A third difference concerns the problem of value discovery. With singleton demand, bidders have only one allocation to evaluate, but in the general case the exponentially larger number of al- locations can force a bidder to reduce its valuation activities, which can limit both efficiency and price competition. Because the Vickrey mechanism plays a significant role in both parts of the theory, we begin by studying this mechanism in the next chapter. Auction theory has grown into a huge area of research, and this book reports on only those parts of the theoretical research that are relatively settled and that, in my opinion, have promise to be helpful to auction designers. With these criteria in mind, I have given only light coverage to some of the elegant formal treatments of how auctions perform when there are very many bidders27 as well as much of the recently develop- ing literature about one or more of these topics: auctions with inter- dependent valuations, collusion among bidders, corrupt auctioneers, purchases for resale, and information processing during auctions. Read- ers who wish to follow the frontiers of auction theory are encouraged to read about these subjects in the new auction literature.
27 This research begins with Wilson (1977) and includes Milgrom (1979) and the especially beautiful results by Pesendorfer and Swinkels (1997, 2000) and Swinkels (2001). P1: FCH/FFX P2: FCH/FFX QC: FCH/FFX T1: FCH CB610B-01 CB610-Milgrom-v3 October 6, 2003 11:5
References 33
REFERENCES Bertrand, Joseph (1883). “Theorie´ Mathematique´ de la Richesse Sociale.” Journal des Savants 69: 499–508. Binmore, Ken and Paul Klemperer (2002). “The Biggest Auction Ever: The Sale of the British 3G Telecom Licenses.” Economic Journal 112: C74–C96. Cramton, Peter, John McMillan, Paul Milgrom, Brad Miller, Bridger Mitchell, Daniel Vincent, and Robert Wilson (1997). “Auction Design Enhancements for Non- Combinatorial Auctions.” Report 1a: Market Design, Inc. and Charles River As- sociates, www.market-design.com/files/97cra-auction-design-enhancements- for-non-combinatorial-auctions.pdf. Cremer, Jacques and Richard P.McLean (1985). “Optimal Selling Strategies Under Uncertainty for a Discriminating Monopolist When Demands are Independent.” Econometrica 53(2): 345–361. Dasgupta, Partha and Eric Maskin (2000). “Efficient Auctions.” Quarterly Journal of Economics 95: 341–388. Hazlett, Thomas (1998). “Assigning Property Rights to Radio Spectrum Users: Why Did FCC License Auctions Take 67 Years?” Journal of Law and Economics XLI(2, pt 2): 529–575. Hendricks, Kenneth, Robert Porter, and Charles Wilson (1994). “Auctions for Oil and Gas Leases with an Informed Bidder and a Random Reservation Price.” Econo- metrica 63(1): 1–27. Jehiel, Philippe and Benny Moldovanu (1999). “Resale Markets and the Assignment of Property Rights.” Review of Economic Studies 64(4): 971–991. Kagel, John H. (1995). Auctions: A Survey of Experimental Research. The Hand- book of Experimental Economics. J. H. Kagel and A. E. Roth. Princeton, Princeton University Press. Chapter 7: 501–585. Kwerel, Evan and Alex Felker (1985). “Using Auctions to Select FCC Licensees,” Federal Communications Commission Working Paper 16. McMillan,John(1994).“SellingSpectrumRights.”JournalofEconomicsPerspectives 8: 145–162. Milgrom, Paul (1979). “A Convergence Theorem for Competitive Bidding with Dif- ferential Information.” Econometrica 47: 670–688. Mueller, Milton (1993). “New Zealand’s Revolution in Spectrum Management.” Information Economics and Policy 5: 159–177. Perry, Motty and Philip Reny (2002). “An Efficient Auction.” Econometrica 70(3): 1199–1212. Pesendorfer, Wolfgang and Jeroen Swinkels (1997). “The Loser’s Curse and In- formation Aggregation in Common Value Auctions.” Econometrica 65: 1247– 1281. Pesendorfer, Wolfgang and Jeroen Swinkels (2000). “Efficiency and Information Aggregation in Auctions.” American Economic Review 90(3): 499–525. Roth, Alvin E. (1991). “A Natural Experiment in the Organization of Entry-Level Labor Markets: Regional Markets for New Physicians and Surgeons in the United Kingdom.” American Economic Review 81(3): 415–440. Roth, Alvin E. and Axel Ockenfels (2000). “Last Minute Bidding and the Rules for P1: FCH/FFX P2: FCH/FFX QC: FCH/FFX T1: FCH CB610B-01 CB610-Milgrom-v3 October 6, 2003 11:5
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Ending Second-Price Auctions: Theory and Evidence from a Natural Experiment on the Internet.” NBER Working Paper: 7299. Swinkels, Jeroen (2001). “Efficiency of Large Private Value Auctions.” Econometrica 69(1): 37–68. Vickrey, William (1961). “Counterspeculation, Auctions, and Competitive Sealed Tenders.” Journal of Finance XVI:8–37. Wilson, Robert (1977). “A Bidding Model of Perfect Competition.” Review of Eco- nomic Studies 44: 511–518. P1: FCH/FFX P2: FCH/FFX QC: FCH/FFX T1: FCH CB610B-02part CB610-Milgrom-v3 October 3, 2003 9:49
PART ONE
THE MECHANISM DESIGN APPROACH
The five chapters of Part I apply mechanism design theory and related methods to problems of auction design. We begin with informal de- scriptions of the main concepts of mechanism design theory. Although these descriptions correspond closely to the formal ones, they conceal technical details that are occasionally important, so the mathematical development is indispensable for a full understanding of the theory. Mechanism design theory distinguishes sharply between the appara- tus under the control of the designer, which we call a mechanism, and the world of things that are beyond the designer’s control, which we call the environment. A mechanism consists of rules that govern what the participants are permitted to do and how these permitted actions determine outcomes. An environment comprises three lists: a list of the participants or potential participants, another of the possible outcomes, and another of the participants’ possible types – that is, their capabilities, preferences, information, and beliefs. In a political mechanism model, the participants could be legislators, and an outcome the set of bills that are enacted. Or the participants could be voters, and the outcome a set of elected officials. The mechanism an- alyst might investigate how a particular legislative process affects the likelihood of stalemate or how the electoral system distorts choices by politicians concerned with reelection. In economic mechanism models, the participants could be workers, or the members of a family, or depart- mentalmanagers.Theanalystwouldmodelhowmechanismsdetermine job assignments, the distribution of household chores or the family bud- get, or the levels of funding of departments within a firm. Indeed, the most commonly studied mechanisms in economics are resource alloca- tion mechanisms in which the outcome is an allocation of resources.
35 P1: FCH/FFX P2: FCH/FFX QC: FCH/FFX T1: FCH CB610B-02part CB610-Milgrom-v3 October 3, 2003 9:49
36 The Mechanism Design Approach
Mechanism theory evaluates alternative designs based on their com- parative performance. Formally, performance is the function that maps environments into outcomes. The function “When it rains, we distribute umbrellas; when the sun shines, we distribute bathing suits” gives better performance than the opposite distribution pattern. The goal of mechanism design analysis is to determine what perfor- mance is possible and how mechanisms can best be designed to achieve the designer’s goals. Mechanism design addresses three common ques- tions: Is it possible to achieve a certain kind of performance, for instance a map that picks an efficient allocation for every possible environment in some class?1 What is the complete set of performance functions that are implementable by some mechanism? What mechanism optimizes performance (according to the mechanism designer’s performance criterion)? Mechanism design theory is outcome-oriented. A central assumption of the theory is that people care only about outcomes, not how they are achieved. In the real world, processes sometimes succeed or fail based on whether they are perceived as fair, simple, or open – attributes that are hard to evaluate in a formal model. Setting aside these considerations facilitates a formal but partial analysis. Once that analysis is complete, the omitted issues and criteria can be examined. Two categories of problems plague mechanism designers. Informa- tion problems are the first category. Consider the problem of an airline regulator trying to respond to bad weather around a major airport that requires delaying or canceling some flights. Which flights? The regulator might ask the airlines to cooperate by identifying which flights can be canceled with only moderate disruptions to passengers and the sched- ule, but then airlines that honestly identify those flights will bear most of the cost of cancellations. Canceling flights could even make passenger service problems worse. For example, when flights on large planes are canceled or delayed, some wealthy passengers may hire private jets that use the same runway capacity to serve fewer customers. In this exam- ple, the regulator might be able to alleviate the information problems by
1 In applying the theory, one needs to be cautious about the definition of “efficiency” used in the theory. These formulations focus only on the payoffs to participants in N. If some outcome has value to a participant because it allows him to extract rents through the ap- plication of monopoly power, then the identified allocations are generally not efficient in the economist’s usual sense of Pareto optimality. P1: FCH/FFX P2: FCH/FFX QC: FCH/FFX T1: FCH CB610B-02part CB610-Milgrom-v3 October 3, 2003 9:49
The Mechanism Design Approach 37
paying any airline that voluntarily sacrifices a runway slot and charging a fee to an unscheduled airline seeking an extra slot. In practice, cash compensation may not be allowed. What can be achieved then? What additional performance is possible if cash payments are possible? Problems caused by inadequate information can be found through- out the economy. An architect who requires use of materials of a certain quality may not know that the builder has actually used a less costly and less durable substitute. Black marketeers who conceal their transactions or people who misreport income may thwart a government’s tax system. A business manager may find a system of performance-related pay frus- trated by inaccurate or intentionally distorted performance measures. The second kind of problem facing mechanism designers is a com- mitment problem, in which participants do not trust the designer to keep his promises. For example, suppose the workers in a certain factory are paid a certain amount, called a piece rate, for each unit they produce. The manager of a factory promises not to change a piece rate, regardless of how much workers earn. Suppose the workers believe the manager and increase their output, but it turns out that some workers’ piece rates are much too high relative to others, allowing them to earn much higher incomes. The manager’s superiors and the workers whose piece rates are relatively low are likely to pressure the manager to reduce the higher rates and increase the lower ones. Anticipating such a reaction, the fac- tory workers in the easy jobs may try to make their jobs look hard by limiting their production to avoid a reduction in their piece rates. In this example, the manager’s inability to commit not to change rates reduces the factory output. Both kinds of problems play a role in mechanism design theory and in its application to the economic theory of contracts. We will focus on information problems, however, because these are the most relevant ones for auction theory. They arise for the simple reason that bidders know more about their values than the auctioneer. An auction is a mechanism to allocate resources among a group of bid- ders. An auction model includes three major parts: a description of the potential bidders (and sometimes the seller or sellers), the set of possi- ble resource allocations (describing the number of goods of each type, whether the goods are divisible, and whether there are legal or other restrictions on how the goods may be allocated), and the values of the various resource allocations to each participant. P1: FCH/FFX P2: FCH/FFX QC: FCH/FFX T1: FCH CB610B-02part CB610-Milgrom-v3 October 3, 2003 9:49
38 The Mechanism Design Approach
Values may be determined in subtle ways. For example, when a bottle of fine wine is sold at auction, the winning bidder’s payoff may depend on how much she likes the particular wine, likes the prestige of winning the bottle, or likes keeping the bottle away from a certain competing collector. Losers, too, may care about the outcome, for example because they expect that if a certain friend wins the bottle, he will serve it at an upcoming wine tasting party. The mechanism designer’s problem is to choose the rules of the auction – what bids are allowed, how the resources areallocated,andhowpricesaredetermined–toachievesomeobjective, such as maximizing the seller’s proceeds. Three important early contributions to mechanism design deserve special mention. The next chapter reviews the first of these contribu- tions, William Vickrey’s design of auctions that allocate resources effi- ciently in a wide range of circumstances. The second important contribution was the Vickrey–Mirrlees design of an optimal income tax and welfare system given a utilitarian objective. Vickrey built the basic model, which gave structure to the question. The model incorporated the ideas that individual utility depends on income and leisure, that different people have different opportunities to generate income by sacrificing leisure, that the taxing authority can only observe total income, and that the tax system affects labor supply. The problem was to create a tax-and-transfer system to maximize the total utility of everyone in society. The utilitarian optimal solution would tax those with high earning ability and pay transfers to those with low earning ability, but would be limited by the incentive problem that entails. James Mirrlees later revisited and solved the optimization problem implied by Vickrey’s formulation. Subsequent researchers have often mimicked Mirrlees’ methods. For their contributions to the theories of efficient auctions and optimal taxation, Vickrey and Mirrlees shared the 1996 Nobel Prize in economic science. The third important contribution was the Clarke–Groves analysis of the optimal provision of public goods. For example, a condominium association may need to decide whether to improve its common areas, perhaps by installing a faster elevator in the building, renovating the exterior, or building a children’s playground. Improvements are costly and must be funded out of association funds and by an assessment levied on the association members. In these circumstances, the association board may want to know how much various improvements are worth to P1: FCH/FFX P2: FCH/FFX QC: FCH/FFX T1: FCH CB610B-02part CB610-Milgrom-v3 October 3, 2003 9:49
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its members. Depending on exactly how the information is used and how costsareshared,associationmembersmightbeinclinedtomisstatetheir preferences. Clarke and Groves analyzed how to arrange affairs to make truthful reporting consistent with individual interests. Their methods and conclusions are quite similar to Vickrey’s; we treat the two together in chapter 2. Intheyearsthatfollowed,mechanismdesigntechniqueswereapplied to problems in the public sector, e.g. the optimal state regulation of pub- lic utilities to maximize consumer welfare, and the private sector, e.g. the optimal design of contracts to maximize the welfare of the contract- ing parties. Roger Myerson’s work on designing auctions to maximize revenue was the first to apply mechanism design to auction theory.2
Formalities of the Mechanism Design Model3 The model we shall study has two parts: an environment and a mecha- nism. In the simplest case, an environment is a triple (N,,). The first element of the triple, N ={1,...,n}, is the list of participants (or poten- tial participants) in the mechanism. When it is convenient to include the mechanism designer among the participants, we may instead write N ={0,...,n}. The second element, , is the set of possible outcomes over which the participants and the mechanism designer have prefer- ences. The third element is the most abstract one: = 1 ×···×N is the set of type profilest = (t1,...,tN), which includes a type for each participant. Participant i’s type (ti ) indexes the participant’s information, beliefs, and preferences. For example, we may say that bidder 1 is of type A if the item for sale is worth $100 to that bidder and the bidder believes that the item is worth $150 to bidder 2, and of type B if the item is worth $200 to the bidder and the bidder believes it is worth $175 to bidder 2. The set of types lists all the possibilities that the modeler considers. The type profile and the outcome combine to determine individual payoffs: ui : × → R. Thus, ui (ξ,t) denotes the payoff or utility that participant i gets when the outcome is ξ ∈ and the type profile ist. In much of economic theory, a player’s payoff depends only on the outcome and his own type, but the general formulation allows a broader dependence than that. An example in which payoffs depend on others’
2 See Myerson (1981). 3 The first general mechanism design model was formulated by Hurwicz (1973). P1: FCH/FFX P2: FCH/FFX QC: FCH/FFX T1: FCH CB610B-02part CB610-Milgrom-v3 October 3, 2003 9:49
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types comes from George Akerlof, who shared the 2001 Nobel Prize in economic science. In his famous lemons model of the market for used cars,4 there are two kinds of participants: buyers and sellers. A seller’s type describes the car’s condition, which only the seller knows. A buyer’s utility depends on both the buyer’s tastes and the car’s condition. Market models in which some participant has quality information that affects other participants’ payoffs are called adverse selection models. The name reflects the idea that the selection of cars being sold in this model is not a random cross section of all cars but instead is overweighted by cars that are in bad condition, because owners of bad cars are more eager to sell them. Although the treatment of adverse selection in auction models has a long history,5 the largest part of auction theory sets adverse selection aside to focus on the private-values case, in which each participant’s utilitydependsonlyonitsowntype:ui (ξ,t) = ui (ξ,ti ).Inthiscase,others’ information cannot influence a participant’s ranking of the outcomes in . Except where specifically noted, all the models in this book deal with the private-values case. Most mechanism models assume that participants are uncertain about what other participants know. In Bayesian models, the condi- tional probability distribution π i (t|ti ) describes a participant’s beliefs, which depend on the participant’s own type. Throughout most of this book, we employ the Harsanyi doctrine that the beliefs are derived from a common prior distribution π.6 Although this doctrine is restrictive and rules out certain interesting and realistic phenomena, it does have one important advantage. It rules out betting pathologies, which are mod- els in which participants can make themselves much better off simply by betting against one another based on the differences in their be- liefs.7 The Harsanyi doctrine is popular in mechanism design models,
4 Akerlof (1970). 5 There were auction models with adverse selection even before the pioneering work of Akerlof (1970). See Ortega-Reichert (1968) and Wilson (1969). 6 Harsanyi (1967–1968). 7 Legend has it that the betting pathology was first discovered in the coffee room of the Stanford University economics department, when Professors Joseph Stiglitz and Robert Wilsondisagreedaboutwhetheracertainuncomfortableseatcushionwasstuffedwithfoam or feathers. They agreed to bet $10 on the issue and to cut open the cushion, with the loser to pay for a new cushion. Alas, the department administrator stopped them before they could executetheiragreement.Thepathologyisthatthisagreement,fromwhichbothparticipants P1: FCH/FFX P2: FCH/FFX QC: FCH/FFX T1: FCH CB610B-02part CB610-Milgrom-v3 October 3, 2003 9:49
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because it eliminates such bets and focuses attention on other aspects of the design problem. It is sometimes convenient to write a type profile as t = (ti , t−i ), where t−i lists the types of the participants other than i.A(strategic form) mech- anism is a pair (S,ω) where S = S1 ×···×SN is the set of possible strat- egy profiles (S j is the set of possible strategies of a typical player j) and ω : S → maps strategy profiles to outcomes.8 For each mechanism and each realization t of the type vector, we can define a corresponding strategic form game. The game (N, S, U(·|t)) is a triple consisting of a set of players, a set of strategy profiles, and a payoff function U mapping strategy profiles into payoffs. The argu- ments of the payoff function are strategies, but these matter to the players only insofar as they determine the outcomes that the partici- pants care about: Ui (σ 1,...,σn,t) = ui (ω(σ 1,...,σn),t). If the players are Bayesians, adding the beliefs as described above completes the de- scription of a Bayesian game. Given a mechanism (S, ω), if the game theoretic solution concept forecasts that a particular strategy profile σ = (σ 1(t1),...,σn(tn)) will be played, then one can use that forecast to predict and evaluate the performance of the mechanism. The forecasted outcome will be ξ(t) = ω(σ 1(t1),...,σn(tn)). The function ξ(·) mapping type profiles to outcomes is the performance function corresponding to the mechanism (S, ω). Many game theoretic solution concepts are not single-valued; for example, many games have multiple Nash equilibria. There are several ways to accommodate multiple equilibria, but for part I we focus on
expected to benefit, required the destruction of real resources. It would have been possible to buy a new cushion without destroying the old one first, but that would not have allowed the professors to benefit from the bet. When the Harsanyi doctrine does not hold and parties with the same information nevertheless have different beliefs, side bets like that between Wilson and Stiglitz are quite generally beneficial. According to the no trade theorem (Milgrom and Stokey (1982)), the Harsanyi doctrine precludes mutually beneficial side bets, so adopting the doctrine focuses the analysis on other, more economically plausible aspects of the mechanism design problem. This resolution is unsatisfying, however, because it is contradicted by evidence about human beliefs. Moreover, we will see later that even with the Harsanyi doctrine, side bets still arise in optimal mechanisms when the participants’ types are statistically correlated (Cremer and McLean (1985)). 8 This is a strategic form description of the mechanism. One can also describe a mechanism in extensive form, by completely describing the succession of possible moves (the game tree), the information available to each player when she moves (the information sets), and the outcome that follows each possible sequence of moves. The difference between the two descriptions is potentially significant when one applies an extensive form solution concept, such as sequential equilibrium or perfect equilibrium. P1: FCH/FFX P2: FCH/FFX QC: FCH/FFX T1: FCH CB610B-02part CB610-Milgrom-v3 October 3, 2003 9:49
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the following one. When a game has multiple solutions, we define the augmented mechanism (S, ω, σ ) to be the mechanism plus a selected solution. The idea is that the solution σ represents a recommendation made by the mechanism designer to the participants. If the recommen- dation is consistent with a solution concept that adequately captures the participants’ incentives, then no participant will have any incentive to deviate from the recommendation, and σ is therefore a reasonable prediction of how the participants will behave. When σ is a solution according to some solution concept, we say that the mechanism (S, ω) or the augmented mechanism (S, ω, σ ) implements the performance ξ = ω ◦ σ . In other words, the equilibrium outcome of the mechanism is ξ, which is obtained from the outcome function
ω when each participant plays according to σi . Sometimes, we attach the name of the solution concept, saying that a mechanism imple- ments in dominant strategies or Bayes–Nash implements the particular performance.
The Chapters of Part I We develop the mechanism design approach to auction theory in a series of steps. In chapter 2, we review the Vickrey analysis of auctions and the related Clarke–Groves analysis of public decisions. The Vickrey–Clarke– Groves (VCG) design establishes a useful benchmark with which subse- quent analyses of resource allocation mechanisms must be compared. Chapter 3 introduces the envelope theorem and some of its most important consequences, including Holmstrom’s lemma and Myerson’s lemma, which are incentive theory analogs of the famous demand theory lemmas of Hotelling and Shepard. Using the envelope theorem allows short proofs of many famous results and reveals their close relationship. Among these are the Green–Laffont–Holmstrom theorem that the VCG mechanisms are the only efficient dominant strategy mechanisms, the Myerson–Satterthwaite theorem about the inescapable inefficiencies of bargainingwithincompleteinformation,theJehiel–Moldovanutheorem abouttheimpossibilityofimplementingefficientoutcomeswithadverse selection, the celebrated payoff and revenue equivalence theorems, the Myerson–Riley–Samuelson optimal auctions theorem, and the McAfee– McMillan weak-cartels theorem. Chapter 4 introduces the single crossing properties, the constraint sim- plification theorem, and the ranking lemma. Together, these facilitate P1: FCH/FFX P2: FCH/FFX QC: FCH/FFX T1: FCH CB610B-02part CB610-Milgrom-v3 October 3, 2003 9:49
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analyses of standard auction designs, the characterization of imple- mentable performance functions, the ranking of standard auctions in various different environments, and a fuller development of optimal auction theory. The models explored in chapters 2–4 are simplified by the assump- tions that bidders know their own values and know nothing about oth- ers’ values. In chapter 5, we explore models in which these assumptions no longer hold, including models in which bidders or the seller invest in additional information and conceal or reveal it. A seller can bene- fit in several ways by revealing valuation information. The information can avoid inefficiencies caused by mistaken evaluations, reduce the risk premia that bidders deduct in valuing uncertain assets, and decrease the information rents that bidders earn. All of these changes can increase expected revenues. Chapter 6 sets a larger context for auctions by treating entry deci- sions and post-auction performance. These larger considerations are extremely important in practice: an auction can hardly be considered optimal if no bidders choose to participate or if the winner defaults on his obligations. They also shift the focus of auction design in several im- portant ways. First, when participation is costly, unless enough profitis left for the bidders, they will not choose to participate, damaging both ef- ficiency and revenues. Maximizing efficiency can involve pre-screening of potential bidders, so that only the most qualified incur the cost of learning their types and preparing bids. Pre-screening and other devices can also help ensure that the selected bidder is able to perform. When bidders differ in their qualifications, evaluating bids becomes more com- plicatedaswell,asthesellerbalanceswhethertoacceptahigherbidfrom a weak buyer who may default or a lower bid from a qualified buyer.
REFERENCES Akerlof, George (1970). “The Market for ‘Lemons’: Quality Uncertainty and the Market Mechanism.” Quarterly Journal of Economics 84: 488–500. Cremer, Jacques and Richard P.McLean (1985). “Optimal Selling Strategies under Uncertainty for a Discriminating Monopolist When Demands Are Independent.” Econometrica 53(2): 345–361. Harsanyi, John (1967–1968). “Games with Incomplete Information Played by Bayesian Players (Parts I–III).” Management Science 14: 159–182, 320–334, 486– 502. P1: FCH/FFX P2: FCH/FFX QC: FCH/FFX T1: FCH CB610B-02part CB610-Milgrom-v3 October 3, 2003 9:49
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Hurwicz, Leonid (1973). “The Design of Mechanisms for Resource Allocation.” American Economic Review 63(2): 1–30. Milgrom, Paul and Nancy Stokey (1982). “Information, Trade and Common Knowl- edge.” Journal of Economic Theory 26:17–27. Myerson, Roger B. (1981). “Optimal Auction Design.” Mathematics of Operations Research 6(1): 58–73. Ortega-Reichert, Armando (1968). Models for Competitive Bidding under Uncer- tainty. Stanford, CA: Stanford University. Wilson, Robert (1969). “Competitive Bidding with Disparate Information.” Man- agement Science 15(7): 446–448. P1: FCH/FFX P2: FCH/FFX QC: FCH/FFX T1: FCH CB610B-02 CB610-Milgrom-v3 October 6, 2003 11:0
CHAPTER TWO Vickrey–Clarke–Groves Mechanisms
This chapter describes the important contributions of Vickrey, Clarke, and Groves (VCG) to the theory of mechanism design. Vickrey (1961) analyzed a situation in which bidders compete to buy or sell a collection of goods. Later, Clarke (1971) and Groves (1973) studied the public choice problem, in which agents decide whether to undertake a public project – e.g. construction of a bridge or highway – whose cost must be borne by the agents. This latter analysis formally includes any choice from a finite set. In particular, it includes the Vickrey analysis for the case of discrete assets.Welimitattentioninthischaptertothecaseoffinitechoicesets,to bypass technical issues associated with infinite choice sets, particularly issues associated with the existence of a best choice. The VCG analysis has become an important standard. It is the work by which nearly all other mechanism design work is judged and in terms of which its contribution is assessed. As we will see in later chapters, there are deep and surprising connections between the VCG theory and many other parts of auction theory.
2.1 Formulation We begin the theoretical development in this section by introducing notation and defining direct mechanisms and VCG mechanisms. Thus, let N ={0,...,n} denote the set of participants, with partici- pant 0 being the mechanism operator. Let X denote the set of possible decisions with typical element x. For chapters 2–5, we assume that the set of participants is exogenously given and omit any analysis of the incentives to participate. An outcome is a pair (x, p) describing a deci- sion x and a vector of positive or negative payments p = (p0, p1,...,p n ) by the participants. For example, in a first-price sealed-bid auction, the
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decision x is a vector where xi = 1 if agent i gets the object and 0 other- wise. The associated vector of payments is p, where pi = bi =−p0 if i bids bi and wins, and in that case, pj = 0 for the other bidders. For most of our analysis, we also assume that each participant i values outcomes according to ui ((x, p),t) ≡ vi (x, ti ) − pi , that is, i’s payoff cor- responding to outcome (x, p)isi’s value of the decision x, which depends only on i’s own type ti , minus the payment that i must make. This quasi- linear specification of the utility function plays an indispensable role in the formal analysis of this chapter. The assumption of quasi-linearity implies that bidders are able to make any cash transfers described by the mechanism, that there exists a cash transfer that exactly compensates any individual for any possible change in outcomes, and that redistribut- ing wealth among the participants would not change this compensatory transfer. These assumptions represent better modeling approximations for some situations than for others. For example, if the bidders are firms with ample liquidity, the assumptions might be a very good approxi- mation of reality, but if they are consumers with significant credit con- straints that apply to the transactions, then the assumptions might be an unacceptably bad fit. Recall that “performance” means the function that maps environ- ments into outcomes. Given our assumptions about the two-part de- scription of outcomes, the performance of any mechanism can be also described in two parts. The decision performance maps types into deci- sions x, whereas the transfer performance maps types into payments or transfers. When the decision x allocates goods, we sometimes also call x the allocation performance. The VCG analysis sometimes attempts to achieve efficient perfor- mance subject to the constraint that transfers add up to zero. Given the assumptions described above, a decision x is efficient if it maximizes the vi , i total value i∈N (x t ). For example, in an auction of a single good, the final allocation is efficient if it awards the good to the bidder who values it the most. In the models studied here, by construction, net pay- ments always add up to zero, because the seller (or mechanism designer) receives any sums that the buyers (or bidders) pay. In some publicly run auctions, the design objective is efficiency as defined above, although revenues (the total transfer to the mechanism designer) may also be an important goal. In private-sector auctions, rev- enues are always an important goal and often the only one. P1: FCH/FFX P2: FCH/FFX QC: FCH/FFX T1: FCH CB610B-02 CB610-Milgrom-v3 October 6, 2003 11:0
2.1 Formulation 47
Sometimes, the designer wants to run an auction in which p0 ≡ 0, that is, in which there is never any net transfer to the auction designer. These balanced-budget mechanisms are useful, for example, in regulatory con- texts where the regulator is not authorized to contribute or collect money from the regulated parties. They also arise in the theory of the firm, where the mechanism operator is similarly restricted. As we will see later, there is often a tension in mechanism design between achieving efficient out- comes and ensuring a balanced budget. The VCG mechanisms are incentive-compatible direct mechanisms. i i i Thismeansthat(1)S= andthat(2)thestrategyprofile(σ (t ) = t )i∈N is an equilibrium. In words, the first condition means that each participant is required to report a possible type to the mechanism operator. We will sometimes speak of direct mechanisms as being pairs (x, p), leaving the strategy set implicit. The second condition, incentive compatibility, means that reporting one’s type truthfully is an equilibrium according to whatever solution concept we have chosen. For VCG mechanisms, we focus on dominant-strategy implementation, so the relevant solution concept is that each participant plays a dominant strategy. One appeal of incentive-compatible direct mechanisms is that they spare participants the need for elaborate strategic calculations: truthful reporting serves each participant’s individual interest. Choosing domi- nant strategies as the solution concept, an incentive-compatible direct mechanism is one for which truthful reporting leads to as high a payoff as any other strategy for all possible types of opponents and all possi- ble actions that these opponents may take. For example, as discussed in chapter 1, it is always optimal for a bidder in a second-price sealed-bid auction for a single good to bid his valuation. Moreover, this truthful bidding strategy is the only strategy that is always optimal, so it is a dom- inant strategy. Thus, the second-price auction is a dominant-strategy incentive-compatible direct mechanism. The operator of a VCG mechanism uses the reported types to compute the maximum total value V(X, N,t) and a corresponding total-value- maximizing decision xˆ(X, N,t) as follows: V(X, N,t) = max v j (x, t j ), (2.1) x∈X j∈N xˆ(X, N,t) ∈ arg max v j (x, t j ). (2.2) x∈X j∈N P1: FCH/FFX P2: FCH/FFX QC: FCH/FFX T1: FCH CB610B-02 CB610-Milgrom-v3 October 6, 2003 11:0
48 Vickrey–Clarke–Groves Mechanisms
One might think that such a direct approach would be doomed to failure, because each participant seems to have an incentive to misrep- resent his preferences to influence the decision in his favor. However, the participants’ incentives depend not only on the decision but also on the cash transfer, which is the clever and surprising part of the VCG mechanism. The VCG mechanism eliminates incentives for misreporting by im- posing on each participant the cost of any distortion he causes. The VCG payment for participant i is set so that i’s report cannot affect the total payoff to the set of other parties, N − i. Notice that 0 ∈ N − i, that is, the set includes the mechanism designer whose payoff is the mechanism’s net receipts. With this principle in mind, let us derive a formula for the VCG pay- ments. To capture the effect of i’s report on the outcome, we introduce a hypothetical null report, which corresponds to bidder i reporting that he is indifferent among the possible decisions and cares only about transfers. When i makes the null report, the VCG mechanism optimally chooses the decision xˆ(X, N − i, t−i ). The resulting total value of the de- cision for the set of participants N − i would be V(X, N − i, t−i ), and the mechanism designer might also collect a payment hi (t−i ) from partici- pant i. Thus, if i makes a null report, the total payoff to the participants in set N − i is
− − V(X, N − i, t i ) + hi (t i ).
The VCG mechanism is constructed so that this same amount is the total payoff to those participants regardless of i’s report. Thus, suppose that when the reported type profile is t, i’s payment is pˆi (X, N,t) + hi (t−i ), so that pˆ i (X, N,t)isi’s additional payment over what i would pay if he made the null report. The decision xˆ(X, N,t) generally de- − pends on i’s report, and the total payoff to members of N i is then v j , , , j + i , , + i −i j∈N−i (xˆ(X N t ) t ) pˆ (X N t ) h (t ).Weequatethistotalvalue with the corresponding total value when i makes the null report: